This analysis was prepared by Francisco Arrieta and Jonathan Edwards.

1 Setup

# options(scipen=999) #prevent scientific notation
# options(scipen=-999) #encourage scientific notation
options(scipen=0) #encourage scientific notation neutral?
# kable table layout options
# do not display NAs and only 2 digits
opts <- options(knitr.kable.NA = '') #knitr.table.format = "latex"

# define table styling options
stable <- function(data, digits = 2) {
  knitr::kable(data, digits=digits) |> 
    # kable_styling(c("striped", "condensed"))
    kable_paper(full_width = F)
}

1.1 libraries

# modelling
library(psych) #factor analysis tools (PAF PAF)
library(lavaan) #causal analysis
library(lm.beta) # add standarized regression coeffs

# stats
library(nortest) #Kolmogorov-Smirnov-Test
library(corrplot) #correlation matrix plot
library(olsrr)  #VIF and Tolerance Values
library(pastecs) # provides function stat.desc
library(REdaS) #Bartelett's Test

# plotting & formatting
library(ggplot2) #better graphs
library(patchwork) # provides wrap_plots for multiplotting 
# library(gridExtra) #provides multiplotting functionality
# library(ggpubr) #provides ggarrange for multiplotting (patchwork better though)
library(semPlot) #for visualization of path diagrams (SEM)
library(lavaanPlot) #for visualization of path diagrams (SEM)
# library(rcompanion)   #Histogram and Normal Curve
library(kableExtra) #makes nice tables

# generic
library(dplyr) #useful data manip functions like arrange, distinct, rename etc included in fpp3
library(stringr) # provides string manip functions like str_split_fixed 
library(Hmisc) #describe function that describes features of dataframes
library(data.table) # creating and manipulating datatables
library(knitr) #rmarkdown tools not sure why useful
library(parameters) #get model outputs in table form (good for making tabs)

1.2 Load data

survey <- read.csv("Case Study III_Structural Equation Modeling.csv")
labels <- read.csv("Variables and Labels_Galeries Lafayette.csv")

dim(survey)
## [1] 553  45
# head(labels)

1.3 labels

#Make labels more readable
#create copy of label column without variable code
labels["Category"] <- sub("[^-]*\\s-","",labels[["Label"]]) 
# labels["Category"] <- sub(".*\\s-","",labels[["Label"]])
# labels

#split this new column (category) into category and short label
labels[c("Category","Label_short")] <- str_split_fixed(labels[["Category"]],"\\?\\s\\s|\\s-", n=2)
# labels[20:25,c("Category","Label_short")]
labels[,c("Variable","Category","Label_short")] |>
  stable()
Variable Category Label_short
Im1 What do GLB represent from your point of view Large Assortment
Im2 What do GLB represent from your point of view Assortment Variety
Im3 What do GLB represent from your point of view Artistic Decoration of Sales Area
Im4 What do GLB represent from your point of view Creative Decoration of Sales Area
Im5 What do GLB represent from your point of view Appealing Arrangement of Shop Windows
Im6 What do GLB represent from your point of view France
Im7 What do GLB represent from your point of view French Savoir-vivre
Im8 What do GLB represent from your point of view Expertise in French Traditional Cuisine
Im9 What do GLB represent from your point of view French Fashion
Im10 What do GLB represent from your point of view Gourmet Food
Im11 What do GLB represent from your point of view High-quality Cosmetics
Im12 What do GLB represent from your point of view Luxury brands
Im13 What do GLB represent from your point of view Up tp date Designer Brands
Im14 What do GLB represent from your point of view Gourmet specialities
Im15 What do GLB represent from your point of view Professional Selection of Brands
Im16 What do GLB represent from your point of view Professional Appearance Towards Customers
Im17 What do GLB represent from your point of view Are Trendy
Im18 What do GLB represent from your point of view Are Hip
Im19 What do GLB represent from your point of view Professional Organization
Im20 What do GLB represent from your point of view Relaxing Shopping
Im21 What do GLB represent from your point of view A Great Place to Stroll
Im22 What do GLB represent from your point of view Intimate Shop Atmosphere
C_CR1 CO-CREATION I would like to participate in an expert-workshop to improve the assortment of Galeries Lafayette Berlin.
C_CR2 CO-CREATION I would be available to take part in another survey at Galeries Lafayette Berlin.
C_CR3 CO-CREATION I would like to become a member of a customer group whose opinion is obtained for new products and major changes.
C_CR4 CO-CREATION I would like to participate in planning and designing special events (e.g. fashion show, introduction of new car models) if asked.
C_REP1 REPURCHASE I will continue to be a loyal customer of Galeries Lafayette Berlin.
C_REP2 REPURCHASE I intend to shop at Galeries Lafayette Berlin in the future.
C_REP3 REPURCAHSE I will surely visit Galeries Lafayette Berlin in the future.
COM_A1 AFFECTIVE COMMITMENT How strongly are you attached to Galeries Lafayette Berlin?
COM_A2 AFFECTIVE COMMITMENT How strongly are you emotionally connected to Galeries Lafayette Berlin?
COM_A3 AFFECTIVE COMMITMENT As a customer I feel (close) attached to GL
COM_A4 AFFECTIVE COMMITMENT feel a strong emotional bond toward GLB
SAT_1 SATISFACTION I am very satisfied with Galeries Lafayette Berlin.
SAT_2 SATISFACTION Overall, I am very satisfied with Galeries Lafayette Berlin.
SAT_3 SATISFACTION How satisfied are you with Galeries Lafayette Berlin?
SAT_P1 SATISFACTION EMPLOYEES The employees are capable and professional.
SAT_P2 SATISFACTION EMPLOYEES The employees know best about their products.
SAT_P3 SATISFACTION EMPLOYEES The employees are well-informed.
SAT_P4 SATISFACTION EMPLOYEES The employees are always helpful.
SAT_P5 SATISFACTION EMPLOYEES The employees are willing to respond to my questions in detail.
SAT_P6 SATISFACTION EMPLOYEES The employees are friendly.
TRU_1 TRUST I have the feeling that I can completely rely on GL.
TRU_2 TRUST GLB will always be honest and trustful with me.
TRU_3 TRUST GL will treat me always fair as a customer

1.4 Clean and handle missing data

# # omit all unanswered
# filter_all(survey, all_vars(. != 999))
# filter_all(survey, any_vars(. %in% c(999)))
# 
# filter_all(select(survey,1:22,"SAT_1"), all_vars(. != 999))
# filter_all(select(survey,1:22,"SAT_1"), any_vars(. %in% c(999)))
# 
# filter_all(data_img_EFA, all_vars(. != 999))
# filter_all(ges, any_vars(. %in% c(999)))
# delete variables unused in analysis (see case study instructions): 
survey <- survey |> select(-c("C_CR2", "SAT_P1", "SAT_P2", "SAT_P3", "SAT_P4", "SAT_P5", "SAT_P6", "TRU_1", "TRU_2", "TRU_3"))

# replace missing data (999) with NA
survey <- data.frame(sapply(survey,function(x) ifelse((x==999),NA,as.numeric(x))))

1.5 Explore Data

2 Dimensions by which Galeries Lafayette is perceived

2.1 ROUND 1: Exploratory factor analysis

2.1.1 Variable selection for EFA

# excluded image variables (in the first round of EFA we don't exclude any image variables...)
exclude=c() 

# the full survey data (includes dependent and independent variables) with excluded image variables (in this first round of EFA we don't exclude anything)
survey_excl_img <- survey |> select(-exclude)

# the data we will use for EFA (images)
data_img_EFA <- survey_excl_img[1:(22-length(exclude))]

2.1.2 handle missing data

# delete missing data (delete listwise)
data_img_EFA <- na.omit(data_img_EFA)

dim(survey)
## [1] 553  35
dim(survey_excl_img)
## [1] 553  35
dim(data_img_EFA)
## [1] 385  22

2.1.3 Check adequacy of correlation Matrix

2.1.3.1 correlation matrix

#plot correlation matrix adjusting parameters to see previously identified groupings
corr_matrix <- cor(data_img_EFA)
corrplot(as.matrix(corr_matrix), 
         method = "color", #col = c("white","white","white","white","white", "lightgrey", "darkgrey", "black"),
         order = "hclust", addrect = 10, rect.col="black", # rect.col="red",
         addCoef.col = 'black', number.cex = .5,
         tl.col ="black", 
         tl.cex = 0.80, 
         )

Variables to look out for going forward:

  • Images 9 and 11 are alone

  • Pairs of images: (17,18), cluster exclusively together, have a very high correlation and similar correlation profiles meaning we might only want to keep one of them. Similar comment to a lesser degree for (6,7) and (16,19).

2.1.3.2 Bartlett’s Test

bart_spher(data_img_EFA)
##  Bartlett's Test of Sphericity
## 
## Call: bart_spher(x = data_img_EFA)
## 
##      X2 = 6451.238
##      df = 231
## p-value < 2.22e-16

The Bartlett Test tests the hypothesis that the sample originates from a population, where all variables are uncorrelated. This would not be good for factor analysis, we want this hypothesis to be rejected meaning p-value < 5%.

In our case we see that it is indeed rejected and that the data is not uncorrelated.

2.1.3.3 KMO

KMOTEST=KMOS(data_img_EFA)
print(KMOTEST, sort=T)
## 
## Kaiser-Meyer-Olkin Statistics
## 
## Call: KMOS(x = data_img_EFA)
## 
## Measures of Sampling Adequacy (MSA):
##       Im2       Im6       Im1      Im20      Im14      Im10       Im7       Im4 
## 0.8224640 0.8224827 0.8244624 0.8266391 0.8267452 0.8285789 0.8448231 0.8542604 
##      Im18       Im3      Im17      Im13      Im12      Im22      Im16      Im11 
## 0.8550678 0.8640362 0.8644991 0.8722220 0.8789413 0.8793157 0.9092200 0.9113882 
##      Im21       Im8       Im9      Im19       Im5      Im15 
## 0.9149654 0.9300079 0.9380091 0.9400714 0.9546668 0.9647563 
## 
## KMO-Criterion: 0.8770975

The KMO of 0.8770975 is above 0.6 which indicates the data is well suited for factor anlysis.

2.1.3.4 Anti-image Correlation

MSA_list <- data.table("Item"=names(KMOTEST$MSA), "MSA"=as.numeric(KMOTEST$MSA))

# Sort table
MSA_list<- MSA_list |> 
  setorder(cols = "MSA")

# Display table
MSA_list |> 
  stable() |>
  row_spec(which(MSA_list[,2]<0.5), bold = T, color = "white", background = "#78BE20")
Item MSA
Im2 0.82
Im6 0.82
Im1 0.82
Im20 0.83
Im14 0.83
Im10 0.83
Im7 0.84
Im4 0.85
Im18 0.86
Im3 0.86
Im17 0.86
Im13 0.87
Im12 0.88
Im22 0.88
Im16 0.91
Im11 0.91
Im21 0.91
Im8 0.93
Im9 0.94
Im19 0.94
Im5 0.95
Im15 0.96

Variables with MSA values above 0.5 are suited for factor analysis. Presence of items with low MSA’s (<0.5) could also indicate that an important topic hasn’t been well covered in the questionnaire.

All variables have MSA above 0.5

2.1.4 Select method: PAF

2.1.4.1 Extract factors

EFA_PAF0 <- psych::fa(data_img_EFA, rotate="varimax", scores=TRUE)
# note: by default number of factors = 1 if it is not specified
2.1.4.1.1 Scree plot
#display Scree-plot
plot(EFA_PAF0$e.values,xlab="Factor Number",
     ylab="Eigenvalue",
     main="Scree plot",
     cex.lab=1.2,
     cex.axis=1.2,
     cex.main=1.8,
     col = "#0099F8",
     pch = 19) 
abline(h=1, col = "#7F35B2")

2.1.4.1.2 Kaiser Criterion
EFA_PAF0_kaiser_nb <- length(which(EFA_PAF0$e.values > 1))
EFA_PAF0_kaiser_nb
## [1] 6

The Kaiser criterion suggests we should retain factors with eigenvalues bigger than 1.

There are 6 factors satisfying this condition.

2.1.4.1.3 Total Variance Explained
#calculate total variance (does not change if number of factors change)
EFA_PAF0_EigenValue <- EFA_PAF0$e.values
EFA_PAF0_Variance <- EFA_PAF0_EigenValue / ncol(data_img_EFA) * 100
EFA_PAF0_SumVariance <- cumsum(EFA_PAF0_EigenValue / ncol(data_img_EFA))
EFA_PAF0_Total_Variance_Explained <- cbind("Factor number"=
                                            seq(1, length.out=length(EFA_PAF0_EigenValue[EFA_PAF0_EigenValue>0])),
                                          EigenValue = EFA_PAF0_EigenValue[EFA_PAF0_EigenValue>0],
                                          Variance = EFA_PAF0_Variance[EFA_PAF0_EigenValue>0],
                                          Total_Variance = EFA_PAF0_SumVariance[EFA_PAF0_EigenValue>0])
#display table
EFA_PAF0_Total_Variance_Explained |> 
  stable()
Factor number EigenValue Variance Total_Variance
1 8.98 40.81 0.41
2 2.47 11.21 0.52
3 1.56 7.10 0.59
4 1.46 6.62 0.66
5 1.25 5.67 0.71
6 1.15 5.22 0.77
7 0.81 3.68 0.80
8 0.71 3.23 0.84
9 0.57 2.58 0.86
10 0.46 2.08 0.88
11 0.36 1.64 0.90
12 0.33 1.51 0.91
13 0.29 1.34 0.93
14 0.28 1.29 0.94
15 0.25 1.13 0.95
16 0.23 1.04 0.96
17 0.20 0.92 0.97
18 0.19 0.85 0.98
19 0.16 0.72 0.99
20 0.12 0.53 0.99
21 0.10 0.46 1.00
22 0.08 0.37 1.00

With 6 factors we would explain 76.6310791% of total variance.

With 7 factors we would explain 80.3133487% of total variance.

# test eigenvalue calculation
factorloadings = EFA_PAF0$loadings[,1] # loadings 1st factor (default is nfactors = 1)
Eigenvalue = sum(factorloadings^2)
Eigenvalue
## [1] 8.377985

2.1.4.2 Determine number of factors to retain

# select nb of factors to test
nf = c(5,6,7,8)

2.1.4.3 PAF orthogonal Varimax with n factors

# perform multiple PAFs one for each factor number in selection
EFA_PAFn = list()

i=1
for (n in nf) {
  # EFA_PAFn[[i]] <- n
  EFA_PAFn[[i]] <- psych::fa(data_img_EFA, rotate="varimax", scores=TRUE, nfactors = n)
  i=i+1
}
names(EFA_PAFn) <- nf
2.1.4.3.1 Communalities
#communalities for all selected number of factors

for (i in 1:length(nf)) {

  cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
    EFA_PAFn_communalities <- data.table("Item"=names(EFA_PAFn[[i]]$communality), 
                             "Communality"=as.numeric(EFA_PAFn[[i]]$communality))

    # Sort table
    EFA_PAFn_communalities <- EFA_PAFn_communalities |>
      setorder(cols = "Communality")
    
    # Display table
    kbl <- EFA_PAFn_communalities |> 
              stable() |>
              row_spec(which(EFA_PAFn_communalities[,1]<.3), bold = T, color = "white", background = "#78BE20")
    
    print(kbl)
    cat("\n\n")

    # test communality calculation
    variableloading = EFA_PAFn[[i]]$loadings["Im9",] # loadings 1st variable
    communality = sum(variableloading^2)
    print(paste0("Communality for Im9 =", communality))
    cat("\n")
    
}
Number of factors = 5
Item Communality
Im9 0.41
Im11 0.41
Im18 0.43
Im16 0.46
Im6 0.50
Im19 0.53
Im17 0.54
Im5 0.55
Im15 0.63
Im21 0.65
Im14 0.66
Im10 0.66
Im7 0.69
Im20 0.69
Im12 0.71
Im13 0.72
Im2 0.76
Im8 0.76
Im22 0.81
Im1 0.82
Im3 0.86
Im4 0.92

[1] “Communality for Im9 =0.405600881872799”

Number of factors = 6
Item Communality
Im11 0.45
Im9 0.46
Im16 0.47
Im19 0.52
Im5 0.55
Im18 0.57
Im15 0.63
Im21 0.65
Im17 0.70
Im13 0.70
Im6 0.71
Im12 0.73
Im8 0.74
Im14 0.76
Im2 0.76
Im10 0.78
Im7 0.78
Im20 0.79
Im22 0.79
Im1 0.84
Im3 0.85
Im4 0.92

[1] “Communality for Im9 =0.463399199002637”

Number of factors = 7
Item Communality
Im11 0.44
Im9 0.46
Im16 0.47
Im19 0.53
Im5 0.54
Im15 0.63
Im21 0.65
Im13 0.70
Im18 0.71
Im8 0.72
Im6 0.76
Im2 0.78
Im22 0.79
Im20 0.79
Im14 0.81
Im12 0.83
Im7 0.85
Im1 0.86
Im3 0.86
Im10 0.89
Im17 0.95
Im4 0.97

[1] “Communality for Im9 =0.455673655610471”

Number of factors = 8
Item Communality
Im11 0.45
Im9 0.46
Im5 0.58
Im19 0.62
Im15 0.65
Im21 0.65
Im13 0.70
Im8 0.72
Im18 0.74
Im6 0.76
Im22 0.78
Im16 0.80
Im2 0.81
Im20 0.81
Im12 0.84
Im7 0.85
Im14 0.85
Im3 0.86
Im10 0.90
Im17 0.93
Im1 0.94
Im4 0.97

[1] “Communality for Im9 =0.455554069181416”

Typically we should think about excluding variables with communalities below 0.3.

Based on the above, no variable should be excluded.

2.1.4.3.2 Factor loadings
# loadings for all selected number of factors

for (i in 1:length(nf)) {
  
cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
      # print(EFA_PAFn[[i]]$loadings, cutoff=0.3)
      # print(print_html(model_parameters(EFA_PAFn[[i]], loadings=T, threshold = 0.3, summary=T)))
     kbl <- model_parameters(EFA_PAFn[[i]], loadings=T, threshold = 0.3, summary=T) |>
        stable()
     print(kbl)
      
cat("\n\n")
}
Number of factors = 5
Variable MR2 MR5 MR4 MR1 MR3 Complexity Uniqueness
Im1 0.85 1.31 0.18
Im2 0.83 1.23 0.24
Im3 0.83 1.55 0.14
Im4 0.87 1.46 0.08
Im5 0.64 1.78 0.45
Im6 0.67 1.22 0.50
Im7 0.79 1.20 0.31
Im8 0.84 1.16 0.24
Im9 0.43 0.39 2.76 0.59
Im10 0.75 1.37 0.34
Im11 0.57 1.56 0.59
Im12 0.79 1.27 0.29
Im13 0.77 1.43 0.28
Im14 0.75 1.37 0.34
Im15 0.60 0.33 2.66 0.37
Im16 0.48 0.37 2.77 0.54
Im17 0.38 0.48 3.44 0.46
Im18 0.31 0.43 3.42 0.57
Im19 0.47 0.40 3.37 0.47
Im20 0.78 1.27 0.31
Im21 0.74 1.41 0.35
Im22 0.81 1.51 0.19
Number of factors = 6
Variable MR2 MR5 MR1 MR4 MR3 MR6 Complexity Uniqueness
Im1 0.86 1.30 0.16
Im2 0.83 1.24 0.24
Im3 0.83 1.51 0.15
Im4 0.88 1.42 0.08
Im5 0.64 1.74 0.45
Im6 0.61 0.56 2.12 0.29
Im7 0.72 0.48 1.88 0.22
Im8 0.81 1.25 0.26
Im9 0.35 0.32 0.43 3.51 0.54
Im10 0.80 1.42 0.22
Im11 0.59 1.62 0.55
Im12 0.79 1.34 0.27
Im13 0.73 1.66 0.30
Im14 0.80 1.42 0.24
Im15 0.60 2.76 0.37
Im16 0.48 0.37 2.78 0.53
Im17 0.35 0.31 0.39 0.54 3.66 0.30
Im18 0.35 0.50 3.48 0.43
Im19 0.46 0.40 3.45 0.48
Im20 0.84 1.22 0.21
Im21 0.73 1.43 0.35
Im22 0.79 1.60 0.21
Number of factors = 7
Variable MR5 MR1 MR3 MR2 MR4 MR7 MR6 Complexity Uniqueness
Im1 0.86 1.33 0.14
Im2 0.83 1.27 0.22
Im3 0.83 1.56 0.14
Im4 0.90 1.40 0.03
Im5 0.63 1.85 0.46
Im6 0.83 1.25 0.24
Im7 0.32 0.84 1.44 0.15
Im8 0.63 0.51 2.31 0.28
Im9 0.33 0.45 3.45 0.54
Im10 0.87 1.35 0.11
Im11 0.58 1.72 0.56
Im12 0.85 1.30 0.17
Im13 0.72 1.81 0.30
Im14 0.81 1.50 0.19
Im15 0.59 2.97 0.37
Im16 0.46 0.34 3.44 0.53
Im17 0.84 1.76 0.05
Im18 0.72 1.85 0.29
Im19 0.43 0.37 4.25 0.47
Im20 0.85 1.21 0.21
Im21 0.73 1.44 0.35
Im22 0.78 1.62 0.21
Number of factors = 8
Variable MR1 MR3 MR4 MR7 MR2 MR5 MR6 MR8 Complexity Uniqueness
Im1 0.88 1.45 0.06
Im2 0.82 1.45 0.19
Im3 0.81 1.66 0.14
Im4 0.89 1.47 0.03
Im5 0.65 1.84 0.42
Im6 0.83 1.23 0.24
Im7 0.84 0.30 1.40 0.15
Im8 0.54 0.59 2.56 0.28
Im9 0.33 0.46 3.34 0.54
Im10 0.87 1.41 0.10
Im11 0.58 1.72 0.55
Im12 0.86 1.28 0.16
Im13 0.72 1.78 0.30
Im14 0.83 1.49 0.15
Im15 0.47 0.39 4.81 0.35
Im16 0.77 1.78 0.20
Im17 0.82 1.82 0.07
Im18 0.74 1.76 0.26
Im19 0.30 0.54 3.68 0.38
Im20 0.86 1.20 0.19
Im21 0.73 1.44 0.35
Im22 0.78 1.63 0.22

Factor loadings:

  • Number of items per factor >=3
    • this condition is respected in all cases except when we have 8 factors. We eliminate the 8 factor solution
  • 0.40–0.30–0.20 rule: satisfactory variables (a) load onto their primary factor above 0.40, (b) load onto alter-native factors below 0.30, and (c) demonstrate a difference of 0.20 between their primary and alternative factor loadings.
    • 5 factor solution:
      • this condition is not respected for many variables. We eliminate the 5 factor solution
    • 6 factor solution:
      • this condition is not respected for: Im6, Im7, Im9, (Im11), (Im15), Im16, Im17, Im18, Im19 where Im17 and Im18 are particularly spread out
    • 7 factor solution:
      • this condition is not respected for: Im7, Im8, Im9, Im15, Im16, Im19 where Im16 and Im19 are particularly spread out

Based on the above we will prefer the 7 factor solution but we will no doubt have to exclude some variables.

Looking at the correlation matrix, we saw that variables Im16 and Im19 were highly correlated, we probably only need to exclude one of the two, Im19 has the lower communality of the two so we might consider eliminating that one.

Similarly Im17 and Im18 are highly correlated we might want to eliminate Im18 which has the lowest communality of the two, but they have adequately high loadings in the 7 factor solution so not a priority.

We might also exclude Im9 and Im15 as their communality is on the lower end and their loadings are spread out and quite weak. Also Im9 is problematic in both the 6 and 7 factor solution

Thurstone simple structure criteria:

  • Each row (variable) of the factor pattern matrix should have at least one zero

  • Each column (factor) should have at least r zero elements, and the zeros for one factor should be unique from the zeros for the other factors

  • For every pair of columns (factors), there should be at least r variables with a zero coefficient in one column and a non-zero coefficient in the other

  • When r > 3 every pair of columns (factors), should contain a large proportion of variables with zeros in both columns

  • For every pair of columns (factors), there should be only a small proportion of variables with non-zeros in both columns

2.1.4.4 PAF oblique Promax with n factors

# perform multiple PAFs one for each factor number in selection
EFA_PAFn_obl = list()

i=1
for (n in nf) {
  # EFA_PAFn_obl[[i]] <- n
  EFA_PAFn_obl[[i]] <- psych::fa(data_img_EFA, rotate="promax", scores=TRUE, nfactors = n)
  i=i+1
}
names(EFA_PAFn_obl) <- nf

length(EFA_PAFn_obl)
## [1] 4
2.1.4.4.1 Communalities
#communalities for all selected number of factors

for (i in 1:length(nf)) {

  cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
    EFA_PAFn_obl_communalities <- data.table("Item"=names(EFA_PAFn_obl[[i]]$communality), 
                             "Communality"=as.numeric(EFA_PAFn_obl[[i]]$communality))

    # Sort table
    EFA_PAFn_obl_communalities <- EFA_PAFn_obl_communalities |>
      setorder(cols = "Communality")
    
    # Display table  
    kbl <- EFA_PAFn_obl_communalities |> 
              stable() |> 
              row_spec(which(EFA_PAFn_obl_communalities[,1]<.3), bold = T, color = "white", background = "#78BE20")
    
    print(kbl)
    cat("\n\n")

    # test communality calculation
    variableloading = EFA_PAFn_obl[[i]]$loadings["Im9",] # loadings 1st variable
    communality = sum(variableloading^2)
    print(paste0("Communality for Im9 =", communality))
    cat("\n")
    
}
Number of factors = 5
Item Communality
Im9 0.41
Im11 0.41
Im18 0.43
Im16 0.46
Im6 0.50
Im19 0.53
Im17 0.54
Im5 0.55
Im15 0.63
Im21 0.65
Im14 0.66
Im10 0.66
Im7 0.69
Im20 0.69
Im12 0.71
Im13 0.72
Im2 0.76
Im8 0.76
Im22 0.81
Im1 0.82
Im3 0.86
Im4 0.92

[1] “Communality for Im9 =0.272947652560888”

Number of factors = 6
Item Communality
Im11 0.45
Im9 0.46
Im16 0.47
Im19 0.52
Im5 0.55
Im18 0.57
Im15 0.63
Im21 0.65
Im17 0.70
Im13 0.70
Im6 0.71
Im12 0.73
Im8 0.74
Im14 0.76
Im2 0.76
Im10 0.78
Im7 0.78
Im20 0.79
Im22 0.79
Im1 0.84
Im3 0.85
Im4 0.92

[1] “Communality for Im9 =0.339338697735652”

Number of factors = 7
Item Communality
Im11 0.44
Im9 0.46
Im16 0.47
Im19 0.53
Im5 0.54
Im15 0.63
Im21 0.65
Im13 0.70
Im18 0.71
Im8 0.72
Im6 0.76
Im2 0.78
Im22 0.79
Im20 0.79
Im14 0.81
Im12 0.83
Im7 0.85
Im1 0.86
Im3 0.86
Im10 0.89
Im17 0.95
Im4 0.97

[1] “Communality for Im9 =0.279736181915467”

Number of factors = 8
Item Communality
Im11 0.45
Im9 0.46
Im5 0.58
Im19 0.62
Im15 0.65
Im21 0.65
Im13 0.70
Im8 0.72
Im18 0.74
Im6 0.76
Im22 0.78
Im16 0.80
Im2 0.81
Im20 0.81
Im12 0.84
Im7 0.85
Im14 0.85
Im3 0.86
Im10 0.90
Im17 0.93
Im1 0.94
Im4 0.97

[1] “Communality for Im9 =0.302814913694533”

2.1.4.4.2 Factor loadings
# loadings for all selected number of factors

for (i in 1:length(nf)) {
  
cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
      # print(EFA_PAFn_obl[[i]]$loadings, cutoff=0.3)
      # print(print_html(model_parameters(EFA_PAFn_obl[[i]], loadings=T, threshold = 0.3, summary=T)))
     kbl <- model_parameters(EFA_PAFn_obl[[i]], loadings=T, threshold = 0.3, summary=T) |>
        stable()
     print(kbl)
      
cat("\n\n")
}
Number of factors = 5
Variable MR2 MR5 MR1 MR4 MR3 Complexity Uniqueness
Im1 1.07 1.06 0.18
Im2 1.07 1.07 0.24
Im3 1.00 1.03 0.14
Im4 1.07 1.03 0.08
Im5 0.75 1.03 0.45
Im6 0.73 1.11 0.50
Im7 0.87 1.14 0.31
Im8 0.91 1.01 0.24
Im9 0.36 0.37 2.10 0.59
Im10 0.77 1.17 0.34
Im11 0.69 1.08 0.59
Im12 1.00 1.05 0.29
Im13 0.94 1.03 0.28
Im14 0.77 1.12 0.34
Im15 0.62 1.14 0.37
Im16 0.47 1.79 0.54
Im17 0.42 2.05 0.46
Im18 0.39 1.93 0.57
Im19 0.41 2.02 0.47
Im20 0.86 1.04 0.31
Im21 0.79 1.09 0.35
Im22 0.85 1.02 0.19
Number of factors = 6
Variable MR2 MR5 MR1 MR4 MR3 MR6 Complexity Uniqueness
Im1 1.05 1.06 0.16
Im2 1.03 1.07 0.24
Im3 0.99 1.03 0.15
Im4 1.06 1.05 0.08
Im5 0.74 1.03 0.45
Im6 0.52 0.68 2.22 0.29
Im7 0.65 0.56 2.24 0.22
Im8 0.80 1.05 0.26
Im9 0.48 1.95 0.54
Im10 0.81 1.23 0.22
Im11 0.67 1.15 0.55
Im12 0.91 1.03 0.27
Im13 0.79 1.09 0.30
Im14 0.80 1.18 0.24
Im15 0.60 1.18 0.37
Im16 0.46 1.81 0.53
Im17 0.57 1.97 0.30
Im18 0.54 1.91 0.43
Im19 0.39 2.15 0.48
Im20 0.92 1.07 0.21
Im21 0.76 1.07 0.35
Im22 0.80 1.04 0.21
Number of factors = 7
Variable MR5 MR1 MR2 MR3 MR4 MR7 MR6 Complexity Uniqueness
Im1 1.08 1.06 0.14
Im2 1.05 1.06 0.22
Im3 1.00 1.02 0.14
Im4 1.13 1.04 0.03
Im5 0.72 1.02 0.46
Im6 0.92 1.06 0.24
Im7 0.89 1.06 0.15
Im8 0.62 0.38 1.74 0.28
Im9 0.42 2.10 0.54
Im10 1.04 1.03 0.11
Im11 0.65 1.14 0.56
Im12 1.01 1.04 0.17
Im13 0.78 1.08 0.30
Im14 0.94 1.01 0.19
Im15 0.60 1.17 0.37
Im16 0.40 2.69 0.53
Im17 1.09 1.02 0.05
Im18 0.92 1.02 0.29
Im19 0.33 3.23 0.47
Im20 0.94 1.05 0.21
Im21 0.78 1.06 0.35
Im22 0.81 1.05 0.21
Number of factors = 8
Variable MR1 MR3 MR7 MR2 MR4 MR5 MR6 MR8 Complexity Uniqueness
Im1 1.04 1.01 0.06
Im2 0.95 1.02 0.19
Im3 0.90 1.02 0.14
Im4 1.03 1.03 0.03
Im5 0.72 1.11 0.42
Im6 0.98 1.06 0.24
Im7 0.95 1.05 0.15
Im8 0.43 0.49 2.39 0.28
Im9 0.45 1.92 0.54
Im10 0.97 1.01 0.10
Im11 0.66 1.15 0.55
Im12 1.04 1.05 0.16
Im13 0.79 1.06 0.30
Im14 0.92 1.02 0.15
Im15 0.35 0.36 2.73 0.35
Im16 1.02 1.03 0.20
Im17 0.98 1.01 0.07
Im18 0.90 1.01 0.26
Im19 0.63 1.11 0.38
Im20 0.98 1.09 0.19
Im21 0.79 1.06 0.35
Im22 0.82 1.06 0.22

7 factors: exclude 9, 15, 16, 19 6 factors: exclude 16, 17, 18, 19, 5 factors: exclude 17, 18

2.2 ROUND 2 : Exploratory factor analysis round 2

2.2.1 Variable selection for EFA

# # perform multiple variable selections
# 
# exclude <- list(c("Im1","Im2","Im16", "Im19","Im15","Im9"),
#              c("Im3","Im4","Im9", "Im15","Im11"))
# 
# survey_excl_img2 = list()
# data_img_EFA2 = list()
# 
# for (i in 1:length(exclude)){
#   survey_excl_img2[[i]] <- survey |> select(-exclude[[i]])
#   data_img_EFA2[[i]] <- survey_excl_img2[[i]][1:(22-length(exclude[[i]]))]
#   # print(survey_excl_img2[[i]])
# }
# 
# # survey_excl_img2[[1]]
# # survey_excl_img2[[2]]
# data_img_EFA2[[1]]
# data_img_EFA2[[2]]
# excluded image variables
# candidates for 7 factors: "Im9",("Im11"),"Im15", "Im19", "Im16"
# candidates for 6 factors: "Im16","Im9", "Im19", ("Im11"), ("Im15"), ("Im18"),"Im17", "Im6"

exclude=c("Im9","Im15","Im8") # "Im16", "Im19","Im9","Im11","Im15"

# the full survey data (includes dependent and independent variables) with excluded image variables
survey_excl_img2 <- survey |> select(-exclude) 

# the data we will use for EFA (images)
data_img_EFA2 <- survey_excl_img2[1:(22-length(exclude))]

The excluded variables correspond to the following:

excludedvars <- filter(labels, Variable %in% exclude)[c("Variable","Label_short")] 

excludedvars |>
  stable()
Variable Label_short
Im8 Expertise in French Traditional Cuisine
Im9 French Fashion
Im15 Professional Selection of Brands

2.2.2 handle missing data

# delete missing data
data_img_EFA2 <- na.omit(data_img_EFA2)

dim(survey)
## [1] 553  35
dim(survey_excl_img2)
## [1] 553  32
dim(data_img_EFA2)
## [1] 394  19

2.2.3 Check adequacy of correlation Matrix

2.2.3.1 correlation matrix

#plot correlation matrix adjusting parameters to see previously identified groupings
corr_matrix <- cor(data_img_EFA2)
corrplot(as.matrix(corr_matrix), 
         method = "color", #col = c("white","white","white","white","white", "lightgrey", "darkgrey", "black"),
         order = "hclust", addrect = 10, rect.col="black", # rect.col="red",
         addCoef.col = 'black', number.cex = .5,
         tl.col ="black", 
         tl.cex = 0.80,
         )

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

Variables to look out for going forward: - Images 9 and 11 are alone - Pairs of images: (17,18), cluster exclusively together, have a very high correlation and similar correlation profiles meaning we might only want to keep one of them. Similar comment to a lesser degree for (6,7) and (16,19).

2.2.3.2 Bartlett’s Test

bart_spher(data_img_EFA2)
##  Bartlett's Test of Sphericity
## 
## Call: bart_spher(x = data_img_EFA2)
## 
##      X2 = 5521.451
##      df = 171
## p-value < 2.22e-16

The Bartlett Test tests the hypothesis that the sample originates from a population, where all variables are uncorrelated. This would not be good for factor analysis, we want this hypothesis to be rejected meaning p-value < 5%.

In our case we see that it is indeed rejected and that the data is not uncorrelated.

2.2.3.3 KMO

KMOTEST=KMOS(data_img_EFA2)
print(KMOTEST, sort=T)
## 
## Kaiser-Meyer-Olkin Statistics
## 
## Call: KMOS(x = data_img_EFA2)
## 
## Measures of Sampling Adequacy (MSA):
##       Im6      Im10       Im7      Im14       Im2       Im1      Im20      Im18 
## 0.7471329 0.7642565 0.7744268 0.7843674 0.7950011 0.7961197 0.8143469 0.8323628 
##       Im4      Im17      Im12       Im3      Im13      Im22      Im11      Im16 
## 0.8394241 0.8427407 0.8483413 0.8497470 0.8567410 0.8730869 0.8958364 0.9007282 
##      Im21      Im19       Im5 
## 0.9102753 0.9199391 0.9515411 
## 
## KMO-Criterion: 0.8416876

The KMO of 0.8416876 is above 0.6 which indicates the data is well suited for factor anlysis.

2.2.3.4 Anti-image Correlation

MSA_list <- data.table("Item"=names(KMOTEST$MSA), "MSA"=as.numeric(KMOTEST$MSA))

#Display table
MSA_list<- MSA_list |> 
  setorder(cols = "MSA")
  
MSA_list |> 
  stable() |> 
  row_spec(which(MSA_list[,2]<0.5), bold = T, color = "white", background = "#78BE20")
Item MSA
Im6 0.75
Im10 0.76
Im7 0.77
Im14 0.78
Im2 0.80
Im1 0.80
Im20 0.81
Im18 0.83
Im4 0.84
Im17 0.84
Im12 0.85
Im3 0.85
Im13 0.86
Im22 0.87
Im11 0.90
Im16 0.90
Im21 0.91
Im19 0.92
Im5 0.95

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

Variables with MSA values above 0.5 are suited for factor analysis. Presence of items with low MSA’s (<0.5) could also indicate that an important topic hasn’t been well covered in the questionnaire.

All variables have MSA above 0.5

2.2.4 Select method: PAF

2.2.4.1 Extract factors

EFA_PAF0 <- psych::fa(data_img_EFA2, rotate="varimax", scores=TRUE)
# note: by default number of factors = 1 if it is not specified
2.2.4.1.1 Scree plot
#display Scree-plot
plot(EFA_PAF0$e.values,xlab="Factor Number",
     ylab="Eigenvalue",
     main="Scree plot",
     cex.lab=1.2,
     cex.axis=1.2,
     cex.main=1.8,
     col = "#0099F8",
     pch = 19) 
abline(h=1, col = "#7F35B2")

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

2.2.4.1.2 Kaiser Criterion
EFA_PAF0_kaiser_nb <- length(which(EFA_PAF0$e.values > 1))
EFA_PAF0_kaiser_nb
## [1] 6

The Kaiser criterion suggests we should retain factors with eigenvalues bigger than 1.

There are 6 factors satisfying this condition.

2.2.4.1.3 Total Variance Explained
#calculate total variance (does not change if number of factors change)
EFA_PAF0_EigenValue <- EFA_PAF0$e.values
EFA_PAF0_Variance <- EFA_PAF0_EigenValue / ncol(data_img_EFA2) * 100
EFA_PAF0_SumVariance <- cumsum(EFA_PAF0_EigenValue / ncol(data_img_EFA2))
EFA_PAF0_Total_Variance_Explained <- cbind("Factor number"=
                                            seq(1, length.out=length(EFA_PAF0_EigenValue[EFA_PAF0_EigenValue>0])),
                                          EigenValue = EFA_PAF0_EigenValue[EFA_PAF0_EigenValue>0],
                                          Variance = EFA_PAF0_Variance[EFA_PAF0_EigenValue>0],
                                          Total_Variance = EFA_PAF0_SumVariance[EFA_PAF0_EigenValue>0])
#display table
EFA_PAF0_Total_Variance_Explained |> 
  stable()
Factor number EigenValue Variance Total_Variance
1 7.71 40.57 0.41
2 2.01 10.58 0.51
3 1.54 8.10 0.59
4 1.44 7.58 0.67
5 1.20 6.30 0.73
6 1.06 5.56 0.79
7 0.80 4.21 0.83
8 0.68 3.60 0.87
9 0.50 2.65 0.89
10 0.34 1.81 0.91
11 0.32 1.67 0.93
12 0.30 1.58 0.94
13 0.23 1.20 0.95
14 0.21 1.10 0.97
15 0.19 1.02 0.98
16 0.16 0.84 0.98
17 0.12 0.62 0.99
18 0.11 0.57 1.00
19 0.08 0.43 1.00

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

With 6 factors we would explain 78.6947762% of total variance.

With 7 factors we would explain 82.9058259% of total variance.

# test eigenvalue calculation
factorloadings = EFA_PAF0$loadings[,1] # loadings 1st factor (default is nfactors = 1)
Eigenvalue = sum(factorloadings^2)
Eigenvalue
## [1] 7.109148

2.2.4.2 Determine number of factors to retain

# select nb of factors to test
nf = c(5,6,7,8)

2.2.4.3 PAF orthogonal Varimax with n factors

# perform multiple PAFs one for each factor number in selection
EFA_PAFn = list()

i=1
for (n in nf) {
  # EFA_PAFn[[i]] <- n
  EFA_PAFn[[i]] <- psych::fa(data_img_EFA2, rotate="varimax", scores=TRUE, nfactors = n)
  i=i+1
}
names(EFA_PAFn) <- nf
2.2.4.3.1 Communalities
#communalities for all selected number of factors

for (i in 1:length(nf)) {

  cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")

    EFA_PAFn_communalities <- data.table("Item"=names(EFA_PAFn[[i]]$communality), 
                             "Communality"=as.numeric(EFA_PAFn[[i]]$communality))

    # Sort table
    EFA_PAFn_communalities <- EFA_PAFn_communalities |>
      setorder(cols = "Communality")
    
    # Display table
    kbl <- EFA_PAFn_communalities |> 
              stable() |> 
              row_spec(which(EFA_PAFn_communalities[,1]<.3), bold = T, color = "white", background = "#78BE20")
    
    print(kbl)
    cat("\n\n")

    # test communality calculation
    variableloading = EFA_PAFn[[i]]$loadings["Im6",] # loadings 1st variable
    communality = sum(variableloading^2)
    print(paste0("Communality for Im6 =", communality))
    cat("\n")
    
}
Number of factors = 5
Item Communality
Im11 0.41
Im18 0.42
Im16 0.42
Im6 0.50
Im19 0.51
Im17 0.52
Im5 0.55
Im21 0.64
Im10 0.66
Im7 0.67
Im14 0.70
Im20 0.72
Im13 0.72
Im12 0.75
Im2 0.78
Im22 0.80
Im1 0.84
Im3 0.86
Im4 0.92

[1] “Communality for Im6 =0.496673139129337”

Number of factors = 6
Item Communality
Im16 0.42
Im11 0.45
Im19 0.51
Im5 0.55
Im6 0.64
Im21 0.64
Im18 0.65
Im13 0.69
Im10 0.71
Im7 0.73
Im12 0.74
Im14 0.74
Im20 0.78
Im22 0.79
Im2 0.80
Im17 0.83
Im3 0.86
Im1 0.89
Im4 0.92

[1] “Communality for Im6 =0.638799740882147”

Number of factors = 7
Item Communality
Im16 0.43
Im11 0.43
Im19 0.51
Im5 0.55
Im21 0.64
Im18 0.67
Im13 0.69
Im7 0.72
Im14 0.78
Im22 0.78
Im20 0.79
Im2 0.81
Im3 0.86
Im12 0.87
Im1 0.91
Im6 0.93
Im4 0.97
Im10 0.97
Im17 1.00

[1] “Communality for Im6 =0.931925021212407”

Number of factors = 8
Item Communality
Im11 0.43
Im5 0.58
Im21 0.64
Im16 0.67
Im18 0.68
Im13 0.69
Im7 0.69
Im19 0.70
Im2 0.76
Im14 0.78
Im22 0.78
Im20 0.81
Im3 0.86
Im12 0.87
Im4 0.97
Im6 0.99
Im10 1.00
Im17 1.00
Im1 1.00

[1] “Communality for Im6 =0.990898460352201”

Typically we should think about excluding variables with communalities below 0.3.

Based on the above, no variable should be excluded.

2.2.4.3.2 Factor loadings
# loadings for all selected number of factors

for (i in 1:length(nf)) {
  
cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
      # print(EFA_PAFn[[i]]$loadings, cutoff=0.3)
      # print(print_html(model_parameters(EFA_PAFn[[i]], loadings=T, threshold = 0.3, summary=T)))
     kbl <- model_parameters(EFA_PAFn[[i]], loadings=T, threshold = 0.3, summary=T) |>
        stable()
     print(kbl)
           
cat("\n\n")
}
Number of factors = 5
Variable MR1 MR2 MR4 MR5 MR3 Complexity Uniqueness
Im1 0.85 1.35 0.16
Im2 0.83 1.27 0.22
Im3 0.84 1.44 0.14
Im4 0.89 1.36 0.08
Im5 0.65 1.66 0.45
Im6 0.67 1.23 0.50
Im7 0.78 1.20 0.33
Im10 0.74 1.42 0.34
Im11 0.58 1.49 0.59
Im12 0.82 1.22 0.25
Im13 0.77 1.44 0.28
Im14 0.77 1.39 0.30
Im16 0.40 0.41 3.03 0.58
Im17 0.30 0.43 0.39 3.76 0.48
Im18 0.39 0.32 3.81 0.58
Im19 0.42 0.43 3.53 0.49
Im20 0.80 1.23 0.28
Im21 0.74 1.39 0.36
Im22 0.81 1.49 0.20
Number of factors = 6
Variable MR1 MR2 MR3 MR4 MR5 MR6 Complexity Uniqueness
Im1 0.86 1.40 0.11
Im2 0.83 1.33 0.20
Im3 0.85 1.43 0.14
Im4 0.89 1.35 0.08
Im5 0.65 1.63 0.45
Im6 0.73 1.40 0.36
Im7 0.81 1.24 0.27
Im10 0.70 0.32 1.95 0.29
Im11 0.61 1.46 0.55
Im12 0.80 1.31 0.26
Im13 0.71 1.80 0.31
Im14 0.73 0.32 1.85 0.26
Im16 0.41 0.39 3.16 0.58
Im17 0.74 2.15 0.17
Im18 0.66 2.10 0.35
Im19 0.43 0.39 3.93 0.49
Im20 0.84 1.21 0.22
Im21 0.73 1.43 0.36
Im22 0.79 1.54 0.21
Number of factors = 7
Variable MR1 MR3 MR4 MR5 MR2 MR6 MR7 Complexity Uniqueness
Im1 0.87 1.40 0.09
Im2 0.83 1.35 0.19
Im3 0.84 1.46 0.14
Im4 0.92 1.33 0.03
Im5 0.64 1.73 0.45
Im6 0.93 1.17 0.07
Im7 0.33 0.75 1.62 0.28
Im10 0.92 1.32 0.03
Im11 0.57 1.70 0.57
Im12 0.88 1.23 0.13
Im13 0.72 1.76 0.31
Im14 0.77 0.30 1.67 0.22
Im16 0.38 0.37 3.99 0.57
Im17 0.88 1.60 0.00
Im18 0.70 1.84 0.33
Im19 0.40 0.37 4.52 0.49
Im20 0.85 1.19 0.21
Im21 0.73 1.43 0.36
Im22 0.79 1.55 0.22
Number of factors = 8
Variable MR1 MR3 MR4 MR5 MR2 MR7 MR6 MR8 Complexity Uniqueness
Im1 0.91 1.44 0.00
Im2 0.78 1.56 0.24
Im3 0.82 1.63 0.14
Im4 0.89 1.46 0.03
Im5 0.66 1.78 0.42
Im6 0.96 1.15 0.01
Im7 0.34 0.72 1.69 0.31
Im10 0.93 1.32 0.00
Im11 0.57 1.71 0.57
Im12 0.89 1.22 0.13
Im13 0.72 1.76 0.31
Im14 0.77 1.68 0.22
Im16 0.68 2.03 0.33
Im17 0.88 1.65 0.00
Im18 0.70 1.84 0.32
Im19 0.64 2.71 0.30
Im20 0.86 1.19 0.19
Im21 0.73 1.44 0.36
Im22 0.79 1.56 0.22

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

Factor loadings:

  • Number of items per factor >=3
    • this condition is respected in all cases except when we have 8 factors. We eliminate the 8 factor solution
  • 0.40–0.30–0.20 rule: satisfactory variables (a) load onto their primary factor above 0.40, (b) load onto alter-native factors below 0.30, and (c) demonstrate a difference of 0.20 between their primary and alternative factor loadings.
    • 5 factor solution:
      • this condition is not respected for many variables. We eliminate the 5 factor solution
    • 6 factor solution:
      • this condition is not respected for: Im6, Im7, Im9, (Im11), (Im15), Im16, Im17, Im18, Im19 where Im17 and Im18 are particularly spread out
    • 7 factor solution:
      • this condition is not respected for: Im7, Im8, Im9, Im15, Im16, Im19 where Im16 and Im19 are particularly spread out

Based on the above we will prefer the 7 factor solution but we will no doubt have to exclude some variables, potentially 19 as it also has the lower communality than Im16 and probably also factor 9 as it also has low communality

Thurstone simple structure criteria:

  • Each row (variable) of the factor pattern matrix should have at least one zero

  • Each column (factor) should have at least r zero elements, and the zeros for one factor should be unique from the zeros for the other factors

  • For every pair of columns (factors), there should be at least r variables with a zero coefficient in one column and a non-zero coefficient in the other

  • When r > 3 every pair of columns (factors), should contain a large proportion of variables with zeros in both columns

  • For every pair of columns (factors), there should be only a small proportion of variables with non-zeros in both columns

2.2.4.4 PAF oblique Promax with n factors

# perform multiple PAFs one for each factor number in selection
EFA_PAFn_obl = list()

i=1
for (n in nf) {
  # EFA_PAFn_obl[[i]] <- n
  EFA_PAFn_obl[[i]] <- psych::fa(data_img_EFA2, rotate="promax", scores=TRUE, nfactors = n)
  i=i+1
}
names(EFA_PAFn_obl) <- nf

length(EFA_PAFn_obl)
## [1] 4
2.2.4.4.1 Communalities
#communalities for all selected number of factors

for (i in 1:length(nf)) {

  cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")

    EFA_PAFn_obl_communalities <- data.table("Item"=names(EFA_PAFn_obl[[i]]$communality), 
                             "Communality"=as.numeric(EFA_PAFn_obl[[i]]$communality))

    # Sort table
    EFA_PAFn_obl_communalities <- EFA_PAFn_obl_communalities |>
      setorder(cols = "Communality")
    
    # Display table  
    kbl <- EFA_PAFn_obl_communalities |> 
              stable() |> 
              row_spec(which(EFA_PAFn_obl_communalities[,1]<.3), bold = T, color = "white", background = "#78BE20")
    
    print(kbl)
    cat("\n\n")

    # test communality calculation
    variableloading = EFA_PAFn_obl[[i]]$loadings["Im6",] # loadings 1st variable
    communality = sum(variableloading^2)
    print(paste0("Communality for Im6 =", communality))
    cat("\n")
    
}
Number of factors = 5
Item Communality
Im11 0.41
Im18 0.42
Im16 0.42
Im6 0.50
Im19 0.51
Im17 0.52
Im5 0.55
Im21 0.64
Im10 0.66
Im7 0.67
Im14 0.70
Im20 0.72
Im13 0.72
Im12 0.75
Im2 0.78
Im22 0.80
Im1 0.84
Im3 0.86
Im4 0.92

[1] “Communality for Im6 =0.547199381232095”

Number of factors = 6
Item Communality
Im16 0.42
Im11 0.45
Im19 0.51
Im5 0.55
Im6 0.64
Im21 0.64
Im18 0.65
Im13 0.69
Im10 0.71
Im7 0.73
Im12 0.74
Im14 0.74
Im20 0.78
Im22 0.79
Im2 0.80
Im17 0.83
Im3 0.86
Im1 0.89
Im4 0.92

[1] “Communality for Im6 =0.833006250441541”

Number of factors = 7
Item Communality
Im16 0.43
Im11 0.43
Im19 0.51
Im5 0.55
Im21 0.64
Im18 0.67
Im13 0.69
Im7 0.72
Im14 0.78
Im22 0.78
Im20 0.79
Im2 0.81
Im3 0.86
Im12 0.87
Im1 0.91
Im6 0.93
Im4 0.97
Im10 0.97
Im17 1.00

[1] “Communality for Im6 =0.971113915290573”

Number of factors = 8
Item Communality
Im11 0.43
Im5 0.58
Im21 0.64
Im16 0.67
Im18 0.68
Im13 0.69
Im7 0.69
Im19 0.70
Im2 0.76
Im14 0.78
Im22 0.78
Im20 0.81
Im3 0.86
Im12 0.87
Im4 0.97
Im6 0.99
Im10 1.00
Im17 1.00
Im1 1.00

[1] “Communality for Im6 =1.14058133882705”

2.2.4.4.2 Factor loadings
# loadings for all selected number of factors

test = list()

for (i in 1:length(nf)) {
  
cat("###### Number of factors =", nf[[i]], "{.unnumbered}" ,"\n")
  
      # print(EFA_PAFn_obl[[i]]$loadings, cutoff=0.3)
      # print(print_html(model_parameters(EFA_PAFn_obl[[i]], loadings=T, threshold = 0.3, summary=T)))
     kbl <- model_parameters(EFA_PAFn_obl[[i]], loadings=T, threshold = 0.3, summary=T) |>
        stable()
     print(kbl)
      
cat("\n\n")
}
Number of factors = 5
Variable MR1 MR2 MR5 MR4 MR3 Complexity Uniqueness
Im1 1.06 1.05 0.16
Im2 1.06 1.07 0.22
Im3 1.05 1.04 0.14
Im4 1.11 1.05 0.08
Im5 0.77 1.03 0.45
Im6 0.72 1.12 0.50
Im7 0.85 1.15 0.33
Im10 0.75 1.21 0.34
Im11 0.68 1.09 0.59
Im12 1.00 1.04 0.25
Im13 0.89 1.02 0.28
Im14 0.78 1.14 0.30
Im16 0.33 0.36 2.14 0.58
Im17 0.34 2.56 0.48
Im18 0.31 2.71 0.58
Im19 0.32 0.36 2.28 0.49
Im20 0.88 1.07 0.28
Im21 0.78 1.06 0.36
Im22 0.84 1.02 0.20
Number of factors = 6
Variable MR1 MR2 MR3 MR5 MR4 MR6 Complexity Uniqueness
Im1 1.04 1.05 0.11
Im2 1.00 1.04 0.20
Im3 1.02 1.05 0.14
Im4 1.08 1.05 0.08
Im5 0.76 1.03 0.45
Im6 0.84 1.39 0.36
Im7 0.91 1.18 0.27
Im10 0.68 1.69 0.29
Im11 0.67 1.05 0.55
Im12 0.91 1.06 0.26
Im13 0.77 1.22 0.31
Im14 0.71 1.55 0.26
Im16 0.33 0.33 2.12 0.58
Im17 0.77 1.14 0.17
Im18 0.69 1.14 0.35
Im19 0.33 2.58 0.49
Im20 0.91 1.08 0.22
Im21 0.75 1.06 0.36
Im22 0.80 1.04 0.21
Number of factors = 7
Variable MR1 MR3 MR5 MR4 MR2 MR6 MR7 Complexity Uniqueness
Im1 1.07 1.05 0.09
Im2 1.02 1.03 0.19
Im3 1.01 1.02 0.14
Im4 1.14 1.05 0.03
Im5 0.73 1.02 0.45
Im6 0.98 1.04 0.07
Im7 0.73 1.16 0.28
Im10 1.10 1.03 0.03
Im11 0.60 1.16 0.57
Im12 1.00 1.03 0.13
Im13 0.75 1.11 0.31
Im14 0.87 1.04 0.22
Im16 3.60 0.57
Im17 1.12 1.02 0.00
Im18 0.86 1.03 0.33
Im19 3.47 0.49
Im20 0.94 1.04 0.21
Im21 0.77 1.05 0.36
Im22 0.82 1.03 0.22
Number of factors = 8
Variable MR1 MR3 MR4 MR2 MR5 MR6 MR7 MR8 Complexity Uniqueness
Im1 1.03 1.01 0.00
Im2 0.86 1.01 0.24
Im3 0.90 1.02 0.14
Im4 1.02 1.03 0.03
Im5 0.73 1.09 0.42
Im6 1.06 1.03 0.01
Im7 0.72 1.14 0.31
Im10 1.05 1.02 0.00
Im11 0.60 1.15 0.57
Im12 1.02 1.04 0.13
Im13 0.75 1.09 0.31
Im14 0.82 1.03 0.22
Im16 0.80 1.02 0.33
Im17 1.03 1.01 0.00
Im18 0.81 1.00 0.32
Im19 0.72 1.04 0.30
Im20 0.97 1.09 0.19
Im21 0.77 1.05 0.36
Im22 0.82 1.04 0.22

Here we have excluded variables: Im8, Im9, Im15 corresponding to Expertise in French Traditional Cuisine, French Fashion, Professional Selection of Brands respectively

2.3 Confirmatory factor analysis

We test whether the constructs found in the exploratory phase adequately describe what is going on.

2.3.1 6 factor model

# no excluded variables
CFA_model_img_6f <- "
DECO =~ Im3 + Im4 + Im5
FRENCH =~ Im6 + Im7 + Im8 + Im10 + Im14
ATMOS =~ Im20 + Im21 + Im22
QUAL =~ Im11 + Im12 + Im13
CHOICE =~ Im1 + Im2 + Im15 + Im16 + Im19
BRAND =~ Im17 + Im18 + Im9
"

# # excluded variables: Im9, Im15, Im16, Im19
# CFA_model_img_6f <- "
# QUAL =~ Im11 + Im12 + Im13
# FRENCH =~ Im6 + Im7 + Im8 + Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# DECO =~ Im3 + Im4 + Im5
# CHOICE =~ Im1 + Im2
# BRAND =~ Im17 + Im18
# "

# # excluded variables: Im9, Im15, Im16, Im19, Im8
# CFA_model_img_6f <- "
# QUAL =~ Im11 + Im12 + Im13
# FRENCH =~ Im6 + Im7 + Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# DECO =~ Im3 + Im4 + Im5
# CHOICE =~ Im1 + Im2
# BRAND =~ Im17 + Im18
# "

# CFA_fit_img <- cfa(CFA_model_img_6f, data=data_img_EFA, missing="ML")
CFA_fit_img_6f <- cfa(CFA_model_img_6f, data=survey, missing="ML")

summary(CFA_fit_img_6f, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 110 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        81
## 
##   Number of observations                           553
##   Number of missing patterns                        87
## 
## Model Test User Model:
##                                                       
##   Test statistic                              1442.584
##   Degrees of freedom                               194
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              8838.959
##   Degrees of freedom                               231
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.855
##   Tucker-Lewis Index (TLI)                       0.827
##                                                       
##   Robust Comparative Fit Index (CFI)             0.854
##   Robust Tucker-Lewis Index (TLI)                0.827
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -15479.697
##   Loglikelihood unrestricted model (H1)     -14758.405
##                                                       
##   Akaike (AIC)                               31121.394
##   Bayesian (BIC)                             31470.938
##   Sample-size adjusted Bayesian (SABIC)      31213.808
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.108
##   90 Percent confidence interval - lower         0.103
##   90 Percent confidence interval - upper         0.113
##   P-value H_0: RMSEA <= 0.050                    0.000
##   P-value H_0: RMSEA >= 0.080                    1.000
##                                                       
##   Robust RMSEA                                   0.110
##   90 Percent confidence interval - lower         0.105
##   90 Percent confidence interval - upper         0.116
##   P-value H_0: Robust RMSEA <= 0.050             0.000
##   P-value H_0: Robust RMSEA >= 0.080             1.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.092
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO =~                                                               
##     Im3               1.000                               1.236    0.936
##     Im4               1.057    0.025   42.551    0.000    1.307    0.970
##     Im5               0.820    0.034   23.857    0.000    1.013    0.761
##   FRENCH =~                                                             
##     Im6               1.000                               0.642    0.535
##     Im7               1.219    0.106   11.525    0.000    0.783    0.644
##     Im8               1.244    0.099   12.567    0.000    0.799    0.755
##     Im10              1.251    0.095   13.133    0.000    0.803    0.914
##     Im14              1.244    0.095   13.142    0.000    0.799    0.929
##   ATMOS =~                                                              
##     Im20              1.000                               1.268    0.848
##     Im21              0.848    0.041   20.879    0.000    1.075    0.785
##     Im22              1.053    0.046   22.697    0.000    1.335    0.873
##   QUAL =~                                                               
##     Im11              1.000                               0.703    0.615
##     Im12              1.410    0.094   15.050    0.000    0.991    0.872
##     Im13              1.465    0.105   13.982    0.000    1.030    0.855
##   CHOICE =~                                                             
##     Im1               1.000                               1.232    0.926
##     Im2               0.942    0.027   34.780    0.000    1.160    0.902
##     Im15              0.720    0.036   20.097    0.000    0.887    0.740
##     Im16              0.567    0.041   13.849    0.000    0.699    0.579
##     Im19              0.540    0.038   14.310    0.000    0.666    0.592
##   BRAND =~                                                              
##     Im17              1.000                               1.184    0.952
##     Im18              1.025    0.038   27.280    0.000    1.214    0.868
##     Im9               0.540    0.047   11.386    0.000    0.640    0.474
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO ~~                                                               
##     FRENCH            0.344    0.046    7.416    0.000    0.434    0.434
##     ATMOS             0.730    0.082    8.901    0.000    0.466    0.466
##     QUAL              0.409    0.051    8.040    0.000    0.471    0.471
##     CHOICE            0.782    0.079    9.940    0.000    0.514    0.514
##     BRAND             0.778    0.076   10.268    0.000    0.531    0.531
##   FRENCH ~~                                                             
##     ATMOS             0.265    0.045    5.888    0.000    0.326    0.326
##     QUAL              0.206    0.030    6.812    0.000    0.456    0.456
##     CHOICE            0.299    0.044    6.737    0.000    0.378    0.378
##     BRAND             0.277    0.042    6.547    0.000    0.364    0.364
##   ATMOS ~~                                                              
##     QUAL              0.373    0.053    7.021    0.000    0.419    0.419
##     CHOICE            0.767    0.083    9.216    0.000    0.491    0.491
##     BRAND             0.792    0.081    9.797    0.000    0.528    0.528
##   QUAL ~~                                                               
##     CHOICE            0.460    0.054    8.585    0.000    0.531    0.531
##     BRAND             0.483    0.053    9.113    0.000    0.581    0.581
##   CHOICE ~~                                                             
##     BRAND             0.864    0.078   11.040    0.000    0.592    0.592
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               4.994    0.056   88.528    0.000    4.994    3.785
##    .Im4               4.997    0.057   86.919    0.000    4.997    3.709
##    .Im5               5.034    0.057   87.796    0.000    5.034    3.785
##    .Im6               5.824    0.051  113.338    0.000    5.824    4.849
##    .Im7               5.751    0.053  109.497    0.000    5.751    4.727
##    .Im8               6.000    0.045  133.008    0.000    6.000    5.671
##    .Im10              6.100    0.037  163.041    0.000    6.100    6.945
##    .Im14              6.139    0.037  166.909    0.000    6.139    7.138
##    .Im20              4.672    0.064   73.182    0.000    4.672    3.124
##    .Im21              5.139    0.058   87.973    0.000    5.139    3.751
##    .Im22              4.278    0.065   65.391    0.000    4.278    2.798
##    .Im11              5.653    0.049  115.277    0.000    5.653    4.943
##    .Im12              5.666    0.049  116.095    0.000    5.666    4.983
##    .Im13              5.448    0.052  105.630    0.000    5.448    4.525
##    .Im1               4.792    0.057   84.316    0.000    4.792    3.601
##    .Im2               4.861    0.055   88.357    0.000    4.861    3.779
##    .Im15              5.090    0.051   99.219    0.000    5.090    4.246
##    .Im16              5.130    0.052   98.387    0.000    5.130    4.251
##    .Im19              5.146    0.048  106.829    0.000    5.146    4.578
##    .Im17              5.025    0.053   94.490    0.000    5.025    4.038
##    .Im18              4.595    0.060   76.460    0.000    4.595    3.286
##    .Im9               5.075    0.058   87.318    0.000    5.075    3.757
##     DECO              0.000                               0.000    0.000
##     FRENCH            0.000                               0.000    0.000
##     ATMOS             0.000                               0.000    0.000
##     QUAL              0.000                               0.000    0.000
##     CHOICE            0.000                               0.000    0.000
##     BRAND             0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               0.215    0.024    8.776    0.000    0.215    0.123
##    .Im4               0.108    0.024    4.439    0.000    0.108    0.060
##    .Im5               0.744    0.049   15.192    0.000    0.744    0.421
##    .Im6               1.030    0.066   15.665    0.000    1.030    0.714
##    .Im7               0.867    0.059   14.739    0.000    0.867    0.586
##    .Im8               0.482    0.034   14.159    0.000    0.482    0.430
##    .Im10              0.127    0.013    9.840    0.000    0.127    0.164
##    .Im14              0.101    0.012    8.329    0.000    0.101    0.137
##    .Im20              0.629    0.061   10.379    0.000    0.629    0.281
##    .Im21              0.721    0.057   12.668    0.000    0.721    0.384
##    .Im22              0.554    0.063    8.781    0.000    0.554    0.237
##    .Im11              0.814    0.055   14.805    0.000    0.814    0.622
##    .Im12              0.310    0.039    7.857    0.000    0.310    0.240
##    .Im13              0.390    0.044    8.777    0.000    0.390    0.269
##    .Im1               0.253    0.035    7.312    0.000    0.253    0.143
##    .Im2               0.308    0.033    9.422    0.000    0.308    0.186
##    .Im15              0.651    0.048   13.673    0.000    0.651    0.453
##    .Im16              0.968    0.065   14.927    0.000    0.968    0.664
##    .Im19              0.820    0.055   14.954    0.000    0.820    0.649
##    .Im17              0.146    0.036    4.039    0.000    0.146    0.094
##    .Im18              0.482    0.046   10.494    0.000    0.482    0.247
##    .Im9               1.415    0.089   15.908    0.000    1.415    0.776
##     DECO              1.527    0.107   14.308    0.000    1.000    1.000
##     FRENCH            0.412    0.064    6.422    0.000    1.000    1.000
##     ATMOS             1.608    0.138   11.683    0.000    1.000    1.000
##     QUAL              0.494    0.067    7.365    0.000    1.000    1.000
##     CHOICE            1.518    0.110   13.746    0.000    1.000    1.000
##     BRAND             1.402    0.100   14.036    0.000    1.000    1.000

2.3.1.1 Discussion of global fit measures

Chi square: p-value > 0.05

RMSEA RMSEA <= 0.05 Good fit 0.05 < RMSEA <= 0.08 Acceptable fit 0.08 < RMSEA <= 0.10 Bad fit RMSEA > 0.1 Unacceptable fit

CFI CFI < 0.90 definitely reject model 0.90 < CFI < 0.95 high underrejection rates for misspecified models CFI > 0.95 accept model

2.3.1.2 local fit measures

# semPaths(CFA_fit_img_6f, what = "path", whatLabels = "std", style = "mx",
#          rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
#          nCharNodes = 7,shapeMan = "rectangle", sizeMan = 8, sizeMan2 = 5, 
#          curvePivot=TRUE, edge.label.cex = 1.2, edge.color = "skyblue4")


semPaths(CFA_fit_img_6f, what = "path", whatLabels = "std", style = "mx",
         rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
         nCharNodes = 7,shapeMan = "rectangle", 
         sizeMan = 4, sizeMan2 = 3, sizeInt = 2, sizeLat = 6, asize = 1.5,
         curvePivot=TRUE, edge.label.cex = .8, edge.color = "skyblue4"
         )

lambda = inspect(CFA_fit_img_6f, what="std")$lambda
theta = inspect(CFA_fit_img_6f, what="std")$theta

# create lambda matrix with ones instead of std.all
ones <- lambda
ones[ones>0] <- 1

# a matrix with dimensions of lambda matrix but with lambdas replaced by thetas
theta_lb <- theta %*% ones
2.3.1.2.1 Indicator reliability criterion
# calculate indicator reliabilities (should be larger than 0.4)
indicrel <- lambda^2/(lambda^2 + theta_lb)
# indicrel

# replace all values satisfying condition with NaN for visibility
indicrel_fail <- indicrel
indicrel_fail[indicrel_fail>.4] <- NaN
indicrel_fail
##      DECO FRENCH ATMOS  QUAL CHOICE BRAND
## Im3   NaN    NaN   NaN   NaN    NaN   NaN
## Im4   NaN    NaN   NaN   NaN    NaN   NaN
## Im5   NaN    NaN   NaN   NaN    NaN   NaN
## Im6   NaN  0.286   NaN   NaN    NaN   NaN
## Im7   NaN    NaN   NaN   NaN    NaN   NaN
## Im8   NaN    NaN   NaN   NaN    NaN   NaN
## Im10  NaN    NaN   NaN   NaN    NaN   NaN
## Im14  NaN    NaN   NaN   NaN    NaN   NaN
## Im20  NaN    NaN   NaN   NaN    NaN   NaN
## Im21  NaN    NaN   NaN   NaN    NaN   NaN
## Im22  NaN    NaN   NaN   NaN    NaN   NaN
## Im11  NaN    NaN   NaN 0.378    NaN   NaN
## Im12  NaN    NaN   NaN   NaN    NaN   NaN
## Im13  NaN    NaN   NaN   NaN    NaN   NaN
## Im1   NaN    NaN   NaN   NaN    NaN   NaN
## Im2   NaN    NaN   NaN   NaN    NaN   NaN
## Im15  NaN    NaN   NaN   NaN    NaN   NaN
## Im16  NaN    NaN   NaN   NaN  0.336   NaN
## Im19  NaN    NaN   NaN   NaN  0.351   NaN
## Im17  NaN    NaN   NaN   NaN    NaN   NaN
## Im18  NaN    NaN   NaN   NaN    NaN   NaN
## Im9   NaN    NaN   NaN   NaN    NaN 0.224
2.3.1.2.2 Construct reliability criterion
# calculate construct reliability (should be above .6)
constrrel <- (t(lambda) %*% ones)^2 / ((t(lambda) %*% ones)^2 + t(theta_lb) %*% ones )
# constrrel

# replace all values satisfying condition with NaN for visibility
constrrel_fail <- constrrel
constrrel_fail[constrrel_fail>.6] <- NaN
constrrel_fail
##        DECO FRENCH ATMOS QUAL CHOICE BRAND
## DECO    NaN    NaN   NaN  NaN    NaN   NaN
## FRENCH  NaN    NaN   NaN  NaN    NaN   NaN
## ATMOS   NaN    NaN   NaN  NaN    NaN   NaN
## QUAL    NaN    NaN   NaN  NaN    NaN   NaN
## CHOICE  NaN    NaN   NaN  NaN    NaN   NaN
## BRAND   NaN    NaN   NaN  NaN    NaN   NaN
2.3.1.2.3 Average Variance Extracted criterion
# calculate Average Variance Extracted (should be above .5)
AVE <- (t(lambda) %*% lambda) / (t(lambda) %*% lambda + t(theta_lb) %*% ones )
# avgvar

# replace all values satisfying condition with NaN for visibility
AVE_fail <- AVE
AVE_fail[AVE_fail>.5] <- NaN
AVE_fail
##        DECO FRENCH ATMOS QUAL CHOICE BRAND
## DECO    NaN    NaN   NaN  NaN    NaN   NaN
## FRENCH  NaN    NaN   NaN  NaN    NaN   NaN
## ATMOS   NaN    NaN   NaN  NaN    NaN   NaN
## QUAL    NaN    NaN   NaN  NaN    NaN   NaN
## CHOICE  NaN    NaN   NaN  NaN    NaN   NaN
## BRAND   NaN    NaN   NaN  NaN    NaN   NaN
2.3.1.2.4 Fornell-Larcker Criteria
# correlations between constructs (factors...) should be lower than .7
psi = inspect(CFA_fit_img_6f, what="std")$psi
psi_fail <- psi
psi_fail[psi_fail<.7] <- NaN
psi_fail
##        DECO FRENCH ATMOS QUAL CHOICE BRAND
## DECO      1                               
## FRENCH  NaN      1                        
## ATMOS   NaN    NaN     1                  
## QUAL    NaN    NaN   NaN    1             
## CHOICE  NaN    NaN   NaN  NaN      1      
## BRAND   NaN    NaN   NaN  NaN    NaN     1
# AVE should be higher than squared correlations between constructs

# replace diagonal of psi matrix with AVE values
psi2 <- psi - psi * diag(1,nrow(psi),ncol(psi)) + diag(AVE) * diag(1,nrow(AVE),ncol(AVE))

# create matrix with columns filled with AVE
AVE_full <- AVE
AVE_full[is.na(AVE_full)] <- 0 #replace NAs with 0s
AVE_full <- AVE_full^0 %*% AVE_full # multiply a matrix full of ones with AVE_full to get columns filled with AVE

# substract matrices any psi bigger than AVE will be negative
AVEpsi_fail <- AVE_full - psi2
# AVE_full - psi2
AVEpsi_fail[AVEpsi_fail >= 0] <- NaN

AVE_full
##             DECO    FRENCH     ATMOS      QUAL    CHOICE     BRAND
## DECO   0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
## FRENCH 0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
## ATMOS  0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
## QUAL   0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
## CHOICE 0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
## BRAND  0.7988422 0.5938271 0.6992777 0.6229362 0.5808676 0.6278343
psi
##         DECO FRENCH ATMOS  QUAL CHOICE BRAND
## DECO   1.000                                
## FRENCH 0.434  1.000                         
## ATMOS  0.466  0.326 1.000                   
## QUAL   0.471  0.456 0.419 1.000             
## CHOICE 0.514  0.378 0.491 0.531  1.000      
## BRAND  0.531  0.364 0.528 0.581  0.592 1.000
AVEpsi_fail
##          DECO FRENCH  ATMOS   QUAL CHOICE  BRAND
## DECO        .                                   
## FRENCH    NaN      .                            
## ATMOS     NaN    NaN      .                     
## QUAL      NaN    NaN    NaN      .              
## CHOICE    NaN    NaN    NaN    NaN      .       
## BRAND     NaN    NaN    NaN    NaN -0.011      .

2.3.1.3 Modification indices

arrange(modificationindices(CFA_fit_img_6f),-mi)
##        lhs op  rhs      mi    epc sepc.lv sepc.all sepc.nox
## 1      Im1 ~~  Im2 495.934  0.951   0.951    3.408    3.408
## 2     Im10 ~~ Im14 298.631  0.357   0.357    3.154    3.154
## 3      Im6 ~~  Im7 255.917  0.693   0.693    0.734    0.734
## 4     Im16 ~~ Im19 128.231  0.463   0.463    0.519    0.519
## 5      Im7 ~~  Im8 104.366  0.316   0.316    0.489    0.489
## 6     DECO =~ Im19  76.462  0.344   0.424    0.378    0.378
## 7   FRENCH =~  Im9  73.482  0.783   0.503    0.372    0.372
## 8      Im6 ~~ Im10  55.895 -0.158  -0.158   -0.438   -0.438
## 9      Im1 ~~ Im16  55.410 -0.242  -0.242   -0.489   -0.489
## 10    QUAL =~ Im15  54.596  0.510   0.359    0.299    0.299
## 11     Im6 ~~  Im8  52.130  0.236   0.236    0.335    0.335
## 12     Im7 ~~ Im14  50.063 -0.145  -0.145   -0.491   -0.491
## 13  FRENCH =~ Im19  49.872  0.497   0.319    0.284    0.284
## 14    Im15 ~~ Im16  49.062  0.263   0.263    0.332    0.332
## 15     Im1 ~~ Im19  46.763 -0.204  -0.204   -0.449   -0.449
## 16    QUAL =~  Im9  45.834  0.702   0.494    0.365    0.365
## 17     Im2 ~~ Im19  45.384 -0.198  -0.198   -0.393   -0.393
## 18     Im7 ~~  Im9  45.214  0.340   0.340    0.307    0.307
## 19    DECO =~ Im16  44.633  0.288   0.356    0.295    0.295
## 20   BRAND =~ Im19  43.739  0.300   0.355    0.316    0.316
## 21     Im7 ~~ Im10  41.088 -0.134  -0.134   -0.406   -0.406
## 22    QUAL =~  Im2  38.006 -0.361  -0.254   -0.197   -0.197
## 23   BRAND =~ Im15  37.135  0.254   0.301    0.251    0.251
## 24    Im17 ~~ Im18  36.588  0.762   0.762    2.870    2.870
## 25     Im2 ~~ Im15  35.103 -0.179  -0.179   -0.399   -0.399
## 26    Im15 ~~ Im19  34.736  0.202   0.202    0.277    0.277
## 27    QUAL =~ Im19  33.294  0.434   0.305    0.271    0.271
## 28     Im8 ~~ Im14  32.038 -0.103  -0.103   -0.467   -0.467
## 29     Im6 ~~  Im9  31.342  0.301   0.301    0.249    0.249
## 30     Im1 ~~ Im15  30.586 -0.174  -0.174   -0.429   -0.429
## 31     Im2 ~~ Im16  28.333 -0.170  -0.170   -0.311   -0.311
## 32    DECO =~ Im15  27.803  0.191   0.236    0.197    0.197
## 33     Im6 ~~ Im14  27.360 -0.108  -0.108   -0.335   -0.335
## 34  FRENCH =~ Im16  27.263  0.403   0.259    0.214    0.214
## 35    DECO =~  Im2  26.666 -0.158  -0.195   -0.152   -0.152
## 36   ATMOS =~ Im15  24.669  0.181   0.230    0.192    0.192
## 37   BRAND =~ Im13  23.589  0.232   0.275    0.228    0.228
## 38    DECO =~  Im1  22.892 -0.149  -0.184   -0.138   -0.138
## 39   ATMOS =~  Im2  21.308 -0.142  -0.180   -0.140   -0.140
## 40    Im11 ~~ Im13  21.300 -0.191  -0.191   -0.339   -0.339
## 41   BRAND =~  Im2  21.047 -0.165  -0.195   -0.152   -0.152
## 42  FRENCH =~  Im1  20.125 -0.244  -0.156   -0.118   -0.118
## 43  FRENCH =~ Im15  18.582  0.279   0.179    0.149    0.149
## 44   BRAND =~ Im12  18.526 -0.197  -0.233   -0.205   -0.205
## 45   BRAND =~ Im16  17.630  0.209   0.247    0.205    0.205
## 46   BRAND =~  Im1  16.671 -0.150  -0.177   -0.133   -0.133
## 47   ATMOS =~ Im19  16.384  0.161   0.204    0.181    0.181
## 48   BRAND =~  Im6  14.589  0.161   0.191    0.159    0.159
## 49  CHOICE =~ Im20  14.430 -0.169  -0.208   -0.139   -0.139
## 50    Im14 ~~  Im9  13.864 -0.082  -0.082   -0.217   -0.217
## 51    Im11 ~~ Im12  13.374  0.145   0.145    0.289    0.289
## 52    Im21 ~~ Im22  13.217 -0.270  -0.270   -0.427   -0.427
## 53  CHOICE =~ Im13  13.044  0.152   0.187    0.155    0.155
## 54   ATMOS =~  Im7  12.806  0.134   0.170    0.140    0.140
## 55     Im8 ~~ Im10  12.666 -0.065  -0.065   -0.265   -0.265
## 56  FRENCH =~ Im11  12.345  0.269   0.173    0.151    0.151
## 57  CHOICE =~ Im12  11.541 -0.136  -0.168   -0.148   -0.148
## 58     Im3 ~~  Im4  11.447  0.264   0.264    1.735    1.735
## 59   ATMOS =~ Im16  11.243  0.146   0.185    0.153    0.153
## 60   ATMOS =~ Im12  11.072 -0.115  -0.146   -0.128   -0.128
## 61     Im8 ~~  Im9  10.943  0.125   0.125    0.152    0.152
## 62  FRENCH =~  Im2  10.891 -0.177  -0.114   -0.088   -0.088
## 63     Im7 ~~ Im22  10.253  0.126   0.126    0.182    0.182
## 64    DECO =~  Im9  10.194  0.167   0.207    0.153    0.153
## 65    Im11 ~~  Im9  10.067  0.156   0.156    0.145    0.145
## 66    Im15 ~~  Im9   9.926  0.139   0.139    0.145    0.145
## 67    QUAL =~ Im16   9.231  0.250   0.176    0.146    0.146
## 68     Im8 ~~ Im15   9.159  0.080   0.080    0.144    0.144
## 69   ATMOS =~  Im9   9.070  0.162   0.205    0.152    0.152
## 70    Im13 ~~ Im17   9.051  0.067   0.067    0.281    0.281
## 71     Im6 ~~ Im22   8.985  0.125   0.125    0.166    0.166
## 72    Im20 ~~ Im21   8.930  0.206   0.206    0.306    0.306
## 73   BRAND =~ Im22   8.880  0.147   0.175    0.114    0.114
## 74     Im8 ~~ Im16   8.801  0.095   0.095    0.139    0.139
## 75   BRAND =~ Im20   8.486 -0.141  -0.167   -0.111   -0.111
## 76   BRAND =~  Im7   8.244  0.114   0.135    0.111    0.111
## 77     Im3 ~~  Im1   8.143 -0.045  -0.045   -0.194   -0.194
## 78     Im8 ~~  Im2   7.823 -0.059  -0.059   -0.153   -0.153
## 79    QUAL =~ Im18   7.388 -0.230  -0.161   -0.115   -0.115
## 80    Im10 ~~ Im16   7.338  0.052   0.052    0.149    0.149
## 81    Im13 ~~  Im1   7.263  0.060   0.060    0.191    0.191
## 82     Im4 ~~ Im17   7.109 -0.041  -0.041   -0.322   -0.322
## 83    Im22 ~~ Im12   7.028 -0.078  -0.078   -0.189   -0.189
## 84    Im17 ~~  Im9   7.005 -0.112  -0.112   -0.247   -0.247
## 85    DECO =~  Im6   6.969  0.109   0.134    0.112    0.112
## 86   ATMOS =~ Im11   6.857  0.102   0.130    0.113    0.113
## 87  CHOICE =~  Im9   6.692  0.148   0.182    0.135    0.135
## 88   ATMOS =~  Im6   6.616  0.102   0.130    0.108    0.108
## 89    Im14 ~~  Im2   6.285  0.031   0.031    0.175    0.175
## 90   BRAND =~  Im5   6.246  0.103   0.122    0.092    0.092
## 91   BRAND =~  Im4   6.180 -0.071  -0.084   -0.062   -0.062
## 92    Im22 ~~  Im9   5.999  0.121   0.121    0.136    0.136
## 93    QUAL =~  Im5   5.978  0.169   0.119    0.089    0.089
## 94     Im1 ~~  Im9   5.822 -0.084  -0.084   -0.140   -0.140
## 95    QUAL =~  Im1   5.747 -0.143  -0.100   -0.075   -0.075
## 96  CHOICE =~ Im18   5.737 -0.115  -0.141   -0.101   -0.101
## 97     Im3 ~~  Im5   5.693 -0.084  -0.084   -0.209   -0.209
## 98     Im8 ~~ Im22   5.661  0.070   0.070    0.136    0.136
## 99    DECO =~ Im20   5.578 -0.100  -0.123   -0.082   -0.082
## 100  ATMOS =~  Im5   5.473  0.089   0.112    0.084    0.084
## 101   DECO =~ Im17   5.404 -0.091  -0.112   -0.090   -0.090
## 102   Im11 ~~ Im17   5.351 -0.061  -0.061   -0.177   -0.177
## 103 CHOICE =~  Im5   5.052  0.088   0.108    0.082    0.082
## 104 FRENCH =~ Im13   4.802 -0.157  -0.101   -0.084   -0.084
## 105   DECO =~ Im22   4.756  0.094   0.116    0.076    0.076
## 106   Im22 ~~ Im11   4.722  0.083   0.083    0.124    0.124
## 107 CHOICE =~ Im22   4.719  0.099   0.122    0.080    0.080
## 108   Im13 ~~ Im16   4.625 -0.073  -0.073   -0.118   -0.118
## 109   Im19 ~~ Im17   4.621  0.056   0.056    0.161    0.161
## 110  ATMOS =~ Im10   4.609 -0.038  -0.048   -0.055   -0.055
## 111  BRAND =~ Im10   4.570 -0.040  -0.048   -0.054   -0.054
## 112   Im14 ~~ Im22   4.494 -0.036  -0.036   -0.154   -0.154
## 113  ATMOS =~  Im4   4.422 -0.053  -0.067   -0.050   -0.050
## 114   Im14 ~~ Im15   4.415 -0.032  -0.032   -0.127   -0.127
## 115   Im21 ~~  Im9   4.370 -0.102  -0.102   -0.101   -0.101
## 116   Im10 ~~ Im13   4.362 -0.031  -0.031   -0.138   -0.138
## 117    Im7 ~~ Im15   4.358  0.074   0.074    0.098    0.098
## 118    Im3 ~~ Im22   4.300  0.047   0.047    0.135    0.135
## 119   Im20 ~~  Im1   4.283 -0.056  -0.056   -0.141   -0.141
## 120    Im4 ~~ Im16   4.262  0.048   0.048    0.149    0.149
## 121 FRENCH =~  Im5   4.167  0.144   0.093    0.070    0.070
## 122  ATMOS =~  Im8   4.061  0.057   0.072    0.068    0.068
## 123    Im6 ~~ Im20   4.039 -0.084  -0.084   -0.105   -0.105
## 124   Im10 ~~ Im11   4.002  0.036   0.036    0.111    0.111
## 125   Im20 ~~ Im17   3.990 -0.054  -0.054   -0.177   -0.177
## 126    Im6 ~~ Im11   3.977 -0.083  -0.083   -0.091   -0.091
## 127    Im5 ~~  Im1   3.869  0.051   0.051    0.117    0.117
## 128   Im15 ~~ Im17   3.695  0.045   0.045    0.147    0.147
## 129    Im4 ~~ Im18   3.604  0.035   0.035    0.152    0.152
## 130   Im12 ~~ Im15   3.556  0.050   0.050    0.110    0.110
## 131 FRENCH =~ Im17   3.536 -0.115  -0.074   -0.060   -0.060
## 132   Im14 ~~ Im16   3.474 -0.035  -0.035   -0.110   -0.110
## 133    Im3 ~~ Im15   3.381  0.037   0.037    0.099    0.099
## 134    Im3 ~~ Im17   3.375  0.029   0.029    0.161    0.161
## 135    Im3 ~~ Im19   3.371  0.040   0.040    0.096    0.096
## 136   Im13 ~~ Im15   3.338  0.051   0.051    0.102    0.102
## 137   Im22 ~~  Im1   3.275  0.049   0.049    0.132    0.132
## 138   Im20 ~~ Im13   3.270  0.057   0.057    0.115    0.115
## 139   Im13 ~~  Im2   3.219 -0.040  -0.040   -0.116   -0.116
## 140   Im11 ~~  Im1   3.071 -0.047  -0.047   -0.104   -0.104
## 141 CHOICE =~ Im21   3.048  0.074   0.091    0.066    0.066
## 142   Im10 ~~ Im17   3.000 -0.021  -0.021   -0.157   -0.157
## 143  ATMOS =~  Im1   2.962 -0.054  -0.068   -0.051   -0.051
## 144  ATMOS =~ Im13   2.889  0.062   0.078    0.065    0.065
## 145   Im16 ~~  Im9   2.810  0.089   0.089    0.076    0.076
## 146    Im5 ~~ Im14   2.794  0.028   0.028    0.101    0.101
## 147    Im4 ~~ Im22   2.701 -0.036  -0.036   -0.146   -0.146
## 148 CHOICE =~ Im10   2.670 -0.030  -0.037   -0.042   -0.042
## 149    Im1 ~~ Im17   2.662 -0.031  -0.031   -0.161   -0.161
## 150  ATMOS =~ Im14   2.649 -0.028  -0.035   -0.041   -0.041
## 151    Im4 ~~ Im19   2.526  0.034   0.034    0.113    0.113
## 152    Im4 ~~ Im11   2.492 -0.034  -0.034   -0.115   -0.115
## 153   Im21 ~~ Im18   2.456 -0.050  -0.050   -0.086   -0.086
## 154   Im12 ~~  Im2   2.343 -0.032  -0.032   -0.104   -0.104
## 155    Im6 ~~ Im15   2.316  0.057   0.057    0.070    0.070
## 156   Im12 ~~  Im9   2.302  0.057   0.057    0.086    0.086
## 157   DECO =~ Im12   2.287 -0.054  -0.067   -0.059   -0.059
## 158    Im6 ~~  Im1   2.248 -0.044  -0.044   -0.087   -0.087
## 159    Im3 ~~ Im20   2.237 -0.034  -0.034   -0.091   -0.091
## 160   Im15 ~~ Im18   2.236 -0.043  -0.043   -0.077   -0.077
## 161    Im7 ~~  Im1   2.236 -0.042  -0.042   -0.089   -0.089
## 162    Im6 ~~ Im12   2.164 -0.047  -0.047   -0.083   -0.083
## 163   Im10 ~~ Im12   2.160  0.020   0.020    0.102    0.102
## 164    Im4 ~~  Im2   2.090 -0.022  -0.022   -0.123   -0.123
## 165    Im5 ~~  Im6   1.873 -0.055  -0.055   -0.063   -0.063
## 166    Im4 ~~  Im6   1.846  0.032   0.032    0.096    0.096
## 167   DECO =~ Im10   1.832 -0.025  -0.031   -0.035   -0.035
## 168   Im22 ~~  Im2   1.825 -0.037  -0.037   -0.090   -0.090
## 169   Im10 ~~ Im22   1.781 -0.024  -0.024   -0.090   -0.090
## 170   Im12 ~~ Im17   1.776 -0.028  -0.028   -0.131   -0.131
## 171    Im6 ~~ Im18   1.754  0.047   0.047    0.066    0.066
## 172   Im21 ~~ Im17   1.749  0.035   0.035    0.108    0.108
## 173    Im5 ~~ Im16   1.746 -0.052  -0.052   -0.062   -0.062
## 174    Im3 ~~ Im12   1.732 -0.023  -0.023   -0.087   -0.087
## 175   Im22 ~~ Im13   1.726 -0.041  -0.041   -0.089   -0.089
## 176   Im18 ~~  Im9   1.696 -0.060  -0.060   -0.073   -0.073
## 177  ATMOS =~ Im17   1.682 -0.052  -0.065   -0.053   -0.053
## 178   QUAL =~  Im4   1.672 -0.060  -0.042   -0.031   -0.031
## 179   Im20 ~~ Im19   1.649  0.048   0.048    0.067    0.067
## 180   Im20 ~~  Im2   1.631 -0.035  -0.035   -0.080   -0.080
## 181 FRENCH =~ Im18   1.599 -0.082  -0.052   -0.038   -0.038
## 182    Im4 ~~ Im12   1.546  0.021   0.021    0.113    0.113
## 183 CHOICE =~ Im17   1.544  0.057   0.071    0.057    0.057
## 184    Im6 ~~ Im19   1.483  0.050   0.050    0.055    0.055
## 185   Im12 ~~ Im16   1.481  0.038   0.038    0.070    0.070
## 186    Im7 ~~ Im16   1.479 -0.052  -0.052   -0.056   -0.056
## 187   DECO =~ Im13   1.462  0.045   0.056    0.046    0.046
## 188 FRENCH =~ Im20   1.442 -0.089  -0.057   -0.038   -0.038
## 189   Im12 ~~ Im13   1.442  0.089   0.089    0.255    0.255
## 190  BRAND =~  Im3   1.434  0.033   0.039    0.030    0.030
## 191   Im12 ~~  Im1   1.360 -0.024  -0.024   -0.087   -0.087
## 192    Im3 ~~ Im18   1.301 -0.022  -0.022   -0.067   -0.067
## 193   Im16 ~~ Im17   1.269  0.032   0.032    0.085    0.085
## 194    Im3 ~~ Im11   1.261  0.025   0.025    0.060    0.060
## 195   Im14 ~~ Im17   1.252  0.013   0.013    0.110    0.110
## 196   Im10 ~~ Im19   1.233  0.019   0.019    0.060    0.060
## 197    Im3 ~~ Im10   1.230  0.012   0.012    0.070    0.070
## 198    Im8 ~~ Im18   1.200 -0.027  -0.027   -0.057   -0.057
## 199    Im8 ~~  Im1   1.194 -0.023  -0.023   -0.065   -0.065
## 200   Im21 ~~  Im2   1.185  0.030   0.030    0.063    0.063
## 201   QUAL =~ Im22   1.168 -0.084  -0.059   -0.038   -0.038
## 202 CHOICE =~ Im14   1.154  0.019   0.023    0.027    0.027
## 203   Im10 ~~  Im9   1.119  0.024   0.024    0.057    0.057
## 204   DECO =~ Im18   1.115  0.043   0.053    0.038    0.038
## 205 CHOICE =~  Im4   1.111 -0.028  -0.035   -0.026   -0.026
## 206   Im14 ~~ Im21   1.104  0.018   0.018    0.067    0.067
## 207    Im5 ~~ Im19   1.093 -0.038  -0.038   -0.048   -0.048
## 208    Im4 ~~  Im8   1.089  0.017   0.017    0.076    0.076
## 209   Im22 ~~ Im18   1.078  0.034   0.034    0.065    0.065
## 210   Im22 ~~ Im15   1.067  0.036   0.036    0.059    0.059
## 211   Im13 ~~ Im18   1.031 -0.027  -0.027   -0.062   -0.062
## 212 CHOICE =~  Im7   0.941  0.037   0.046    0.038    0.038
## 213    Im4 ~~ Im10   0.906 -0.010  -0.010   -0.082   -0.082
## 214    Im3 ~~ Im14   0.890 -0.010  -0.010   -0.065   -0.065
## 215   Im10 ~~  Im2   0.889 -0.012  -0.012   -0.061   -0.061
## 216  ATMOS =~  Im3   0.869  0.023   0.029    0.022    0.022
## 217   Im11 ~~ Im16   0.846  0.038   0.038    0.043    0.043
## 218   QUAL =~ Im14   0.840  0.031   0.022    0.026    0.026
## 219   QUAL =~ Im20   0.837  0.069   0.049    0.033    0.033
## 220    Im1 ~~ Im18   0.830  0.021   0.021    0.060    0.060
## 221    Im3 ~~ Im16   0.818  0.022   0.022    0.048    0.048
## 222   DECO =~  Im7   0.814  0.035   0.043    0.036    0.036
## 223    Im2 ~~ Im18   0.811 -0.021  -0.021   -0.054   -0.054
## 224   Im20 ~~ Im12   0.794  0.026   0.026    0.059    0.059
## 225 FRENCH =~ Im22   0.781  0.067   0.043    0.028    0.028
## 226   Im20 ~~ Im16   0.767  0.036   0.036    0.046    0.046
## 227   Im14 ~~ Im18   0.739 -0.012  -0.012   -0.056   -0.056
## 228   Im22 ~~ Im19   0.717 -0.032  -0.032   -0.047   -0.047
## 229   QUAL =~ Im10   0.654 -0.028  -0.020   -0.023   -0.023
## 230   Im14 ~~ Im13   0.649  0.011   0.011    0.057    0.057
## 231   Im14 ~~ Im11   0.639  0.014   0.014    0.048    0.048
## 232 CHOICE =~  Im6   0.612  0.032   0.039    0.033    0.033
## 233   Im19 ~~  Im9   0.595  0.037   0.037    0.035    0.035
## 234  BRAND =~ Im11   0.591 -0.036  -0.043   -0.038   -0.038
## 235    Im6 ~~  Im2   0.577 -0.023  -0.023   -0.040   -0.040
## 236   Im20 ~~ Im15   0.576  0.026   0.026    0.041    0.041
## 237    Im5 ~~ Im22   0.561  0.027   0.027    0.043    0.043
## 238    Im8 ~~ Im19   0.558  0.022   0.022    0.035    0.035
## 239    Im8 ~~ Im21   0.538 -0.022  -0.022   -0.037   -0.037
## 240   Im21 ~~ Im12   0.537  0.021   0.021    0.045    0.045
## 241   Im11 ~~ Im19   0.536  0.028   0.028    0.034    0.034
## 242   Im21 ~~  Im1   0.534  0.020   0.020    0.046    0.046
## 243   QUAL =~  Im7   0.517  0.053   0.037    0.031    0.031
## 244   Im10 ~~ Im18   0.516  0.011   0.011    0.044    0.044
## 245    Im5 ~~ Im11   0.511  0.026   0.026    0.034    0.034
## 246   Im10 ~~ Im20   0.483  0.012   0.012    0.044    0.044
## 247   Im12 ~~ Im18   0.465 -0.017  -0.017   -0.044   -0.044
## 248   Im14 ~~  Im1   0.460  0.008   0.008    0.052    0.052
## 249 FRENCH =~  Im3   0.457 -0.031  -0.020   -0.015   -0.015
## 250   Im20 ~~ Im11   0.453  0.026   0.026    0.036    0.036
## 251    Im3 ~~  Im8   0.442 -0.012  -0.012   -0.036   -0.036
## 252   Im20 ~~ Im18   0.417  0.021   0.021    0.038    0.038
## 253   Im20 ~~ Im22   0.407  0.061   0.061    0.103    0.103
## 254   Im10 ~~  Im1   0.405 -0.008  -0.008   -0.045   -0.045
## 255    Im5 ~~ Im20   0.402  0.023   0.023    0.034    0.034
## 256  BRAND =~ Im14   0.390 -0.011  -0.014   -0.016   -0.016
## 257    Im2 ~~  Im9   0.381 -0.022  -0.022   -0.033   -0.033
## 258   Im21 ~~ Im11   0.373 -0.023  -0.023   -0.030   -0.030
## 259   Im16 ~~ Im18   0.357 -0.021  -0.021   -0.030   -0.030
## 260    Im5 ~~ Im15   0.339 -0.019  -0.019   -0.028   -0.028
## 261   Im21 ~~ Im13   0.328 -0.018  -0.018   -0.034   -0.034
## 262   QUAL =~  Im8   0.320 -0.031  -0.022   -0.021   -0.021
## 263    Im5 ~~ Im21   0.319 -0.021  -0.021   -0.028   -0.028
## 264   DECO =~  Im8   0.317  0.016   0.020    0.019    0.019
## 265    Im7 ~~ Im19   0.317  0.022   0.022    0.026    0.026
## 266    Im4 ~~ Im20   0.315  0.012   0.012    0.047    0.047
## 267   Im11 ~~  Im2   0.310  0.015   0.015    0.030    0.030
## 268    Im5 ~~  Im8   0.298 -0.015  -0.015   -0.026   -0.026
## 269    Im7 ~~ Im20   0.292 -0.021  -0.021   -0.029   -0.029
## 270    Im4 ~~ Im13   0.275 -0.009  -0.009   -0.045   -0.045
## 271    Im3 ~~ Im13   0.274  0.010   0.010    0.033    0.033
## 272    Im8 ~~ Im13   0.252 -0.012  -0.012   -0.028   -0.028
## 273   DECO =~ Im11   0.240  0.019   0.024    0.021    0.021
## 274   Im12 ~~ Im19   0.218  0.013   0.013    0.027    0.027
## 275    Im4 ~~  Im5   0.193  0.017   0.017    0.059    0.059
## 276    Im3 ~~  Im7   0.193 -0.010  -0.010   -0.024   -0.024
## 277    Im7 ~~ Im12   0.191 -0.013  -0.013   -0.025   -0.025
## 278    Im5 ~~ Im17   0.190  0.011   0.011    0.033    0.033
## 279    Im7 ~~ Im21   0.188 -0.017  -0.017   -0.022   -0.022
## 280    Im5 ~~  Im9   0.179  0.020   0.020    0.019    0.019
## 281   Im19 ~~ Im18   0.174 -0.013  -0.013   -0.021   -0.021
## 282   Im21 ~~ Im16   0.170 -0.017  -0.017   -0.020   -0.020
## 283 FRENCH =~  Im4   0.152 -0.018  -0.012   -0.009   -0.009
## 284    Im4 ~~ Im15   0.151 -0.008  -0.008   -0.028   -0.028
## 285   Im21 ~~ Im19   0.149 -0.014  -0.014   -0.019   -0.019
## 286   DECO =~ Im14   0.132 -0.007  -0.008   -0.009   -0.009
## 287    Im4 ~~  Im1   0.125  0.005   0.005    0.033    0.033
## 288    Im3 ~~  Im2   0.124  0.006   0.006    0.022    0.022
## 289    Im2 ~~ Im17   0.116  0.006   0.006    0.031    0.031
## 290   Im10 ~~ Im15   0.108 -0.005  -0.005   -0.018   -0.018
## 291    Im5 ~~ Im10   0.106 -0.006  -0.006   -0.018   -0.018
## 292 FRENCH =~ Im21   0.106  0.023   0.015    0.011    0.011
## 293    Im6 ~~ Im16   0.097 -0.014  -0.014   -0.014   -0.014
## 294 CHOICE =~  Im8   0.094 -0.009  -0.011   -0.010   -0.010
## 295    Im7 ~~ Im11   0.091 -0.012  -0.012   -0.014   -0.014
## 296   Im10 ~~ Im21   0.089 -0.005  -0.005   -0.018   -0.018
## 297   Im11 ~~ Im18   0.089  0.010   0.010    0.015    0.015
## 298    Im5 ~~  Im2   0.087 -0.008  -0.008   -0.016   -0.016
## 299    Im7 ~~ Im18   0.086 -0.010  -0.010   -0.015   -0.015
## 300   QUAL =~  Im6   0.086 -0.023  -0.016   -0.013   -0.013
## 301    Im5 ~~ Im12   0.084  0.008   0.008    0.017    0.017
## 302 CHOICE =~ Im11   0.082 -0.012  -0.015   -0.013   -0.013
## 303    Im8 ~~ Im12   0.076 -0.006  -0.006   -0.016   -0.016
## 304    Im6 ~~ Im13   0.071  0.009   0.009    0.014    0.014
## 305   Im14 ~~ Im20   0.068 -0.004  -0.004   -0.018   -0.018
## 306    Im5 ~~  Im7   0.064  0.010   0.010    0.012    0.012
## 307   Im22 ~~ Im17   0.054  0.006   0.006    0.022    0.022
## 308    Im6 ~~ Im21   0.052 -0.010  -0.010   -0.011   -0.011
## 309   QUAL =~ Im21   0.051  0.016   0.011    0.008    0.008
## 310    Im4 ~~ Im21   0.046  0.005   0.005    0.017    0.017
## 311   Im21 ~~ Im15   0.040 -0.007  -0.007   -0.010   -0.010
## 312   Im14 ~~ Im12   0.039 -0.003  -0.003   -0.015   -0.015
## 313    Im7 ~~ Im13   0.037 -0.006  -0.006   -0.011   -0.011
## 314    Im8 ~~ Im17   0.036 -0.004  -0.004   -0.015   -0.015
## 315   QUAL =~ Im17   0.035 -0.015  -0.011   -0.009   -0.009
## 316  BRAND =~ Im21   0.027 -0.007  -0.009   -0.006   -0.006
## 317   Im14 ~~ Im19   0.027  0.003   0.003    0.010    0.010
## 318   Im20 ~~  Im9   0.021  0.007   0.007    0.008    0.008
## 319    Im7 ~~ Im17   0.021 -0.004  -0.004   -0.011   -0.011
## 320  BRAND =~  Im8   0.020  0.004   0.005    0.005    0.005
## 321 CHOICE =~  Im3   0.018 -0.004  -0.004   -0.003   -0.003
## 322    Im3 ~~  Im9   0.018 -0.004  -0.004   -0.007   -0.007
## 323    Im5 ~~ Im18   0.017  0.004   0.004    0.007    0.007
## 324   DECO =~ Im21   0.017  0.005   0.006    0.005    0.005
## 325    Im8 ~~ Im20   0.014  0.003   0.003    0.006    0.006
## 326    Im4 ~~  Im9   0.014 -0.003  -0.003   -0.008   -0.008
## 327    Im3 ~~ Im21   0.010 -0.002  -0.002   -0.006   -0.006
## 328  ATMOS =~ Im18   0.009  0.004   0.005    0.004    0.004
## 329   Im11 ~~ Im15   0.007 -0.003  -0.003   -0.004   -0.004
## 330    Im7 ~~  Im2   0.006 -0.002  -0.002   -0.004   -0.004
## 331 FRENCH =~ Im12   0.002 -0.003  -0.002   -0.002   -0.002
## 332   Im13 ~~  Im9   0.002  0.002   0.002    0.003    0.003
## 333    Im4 ~~ Im14   0.002  0.000   0.000   -0.004   -0.004
## 334    Im4 ~~  Im7   0.002 -0.001  -0.001   -0.003   -0.003
## 335    Im8 ~~ Im11   0.002  0.001   0.001    0.002    0.002
## 336   Im13 ~~ Im19   0.002  0.001   0.001    0.002    0.002
## 337    Im3 ~~  Im6   0.002  0.001   0.001    0.002    0.002
## 338   Im22 ~~ Im16   0.001  0.001   0.001    0.002    0.002
## 339   QUAL =~  Im3   0.001  0.001   0.001    0.001    0.001
## 340    Im6 ~~ Im17   0.000  0.000   0.000   -0.001   -0.001
## 341    Im5 ~~ Im13   0.000  0.000   0.000    0.000    0.000

2.3.2 7 factor model

Based on the modification indices we create a new model

# # no excluded variables:
# CFA_model_img_7f <- "
# DECO =~ Im3 + Im4 + Im5
# FOOD =~ Im8 + Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# PRODQUAL =~ Im11 + Im12 + Im13
# CHOICE =~ Im1 + Im2 + Im15 + Im16 + Im19
# BRAND =~ Im17 + Im18
# FRENCH =~ Im6 + Im7 + Im9
# "

# MIs indicate separate Im1, Im2 and Im16, Im19 no excluded variables
# Im8 under FRENCH
# exclude Im8 
# exclude Im15
# exclude Im9
CFA_model_img_7f <- "
DECO =~ Im3 + Im4 + Im5
FOOD =~ Im10 + Im14
ATMOS =~ Im20 + Im21 + Im22
PRODQUAL =~ Im11 + Im12 + Im13
CHOICE =~ Im1 + Im2  
PROF =~ Im16 + Im19
BRAND =~ Im17 + Im18
FRENCH =~ Im6 + Im7
"

# # excluded variables: Im9, Im15, Im16, Im19
# CFA_model_img_7f <- "
# QUAL =~ Im11 + Im12 + Im13
# FOOD =~ Im8 + Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# DECO =~ Im3 + Im4 + Im5
# CHOICE =~ Im1 + Im2
# BRAND =~ Im17 + Im18
# FRENCH =~ Im6 + Im7
# "

# # excluded variables: Im9, Im15, Im16, Im19, Im8
# CFA_model_img_7f <- "
# QUAL =~ Im11 + Im12 + Im13
# FOOD =~ Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# DECO =~ Im3 + Im4 + Im5
# CHOICE =~ Im1 + Im2
# BRAND =~ Im17 + Im18
# FRENCH =~ Im6 + Im7
# "

# # 8 factor model: excluded variables: Im9, Im15, Im8, Im11
# CFA_model_img_7f <- "
# QUAL =~ Im12 + Im13
# FOOD =~ Im10 + Im14
# ATMOS =~ Im20 + Im21 + Im22
# DECO =~ Im3 + Im4 + Im5
# CHOICE =~ Im1 + Im2
# BRAND =~ Im17 + Im18
# FRENCH =~ Im6 + Im7
# PROF =~ Im16 + Im19
# "


CFA_fit_img_7f <- cfa(CFA_model_img_7f, data=survey, missing="ML")
summary(CFA_fit_img_7f, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 108 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        85
## 
##   Number of observations                           553
##   Number of missing patterns                        79
## 
## Model Test User Model:
##                                                       
##   Test statistic                               259.047
##   Degrees of freedom                               124
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7474.765
##   Degrees of freedom                               171
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.982
##   Tucker-Lewis Index (TLI)                       0.975
##                                                       
##   Robust Comparative Fit Index (CFI)             0.981
##   Robust Tucker-Lewis Index (TLI)                0.974
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -12973.111
##   Loglikelihood unrestricted model (H1)     -12843.588
##                                                       
##   Akaike (AIC)                               26116.223
##   Bayesian (BIC)                             26483.028
##   Sample-size adjusted Bayesian (SABIC)      26213.200
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.044
##   90 Percent confidence interval - lower         0.037
##   90 Percent confidence interval - upper         0.052
##   P-value H_0: RMSEA <= 0.050                    0.886
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.045
##   90 Percent confidence interval - lower         0.038
##   90 Percent confidence interval - upper         0.053
##   P-value H_0: Robust RMSEA <= 0.050             0.825
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.029
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO =~                                                               
##     Im3               1.000                               1.236    0.937
##     Im4               1.056    0.025   42.717    0.000    1.305    0.969
##     Im5               0.818    0.034   23.815    0.000    1.011    0.760
##   FOOD =~                                                               
##     Im10              1.000                               0.812    0.923
##     Im14              1.015    0.036   28.479    0.000    0.824    0.952
##   ATMOS =~                                                              
##     Im20              1.000                               1.265    0.845
##     Im21              0.849    0.041   20.823    0.000    1.074    0.783
##     Im22              1.060    0.047   22.606    0.000    1.340    0.877
##   PRODQUAL =~                                                           
##     Im11              1.000                               0.703    0.615
##     Im12              1.410    0.094   15.046    0.000    0.991    0.872
##     Im13              1.465    0.105   13.968    0.000    1.030    0.855
##   CHOICE =~                                                             
##     Im1               1.000                               1.305    0.980
##     Im2               0.885    0.033   27.043    0.000    1.155    0.899
##   PROF =~                                                               
##     Im16              1.000                               0.921    0.766
##     Im19              1.046    0.061   17.170    0.000    0.963    0.856
##   BRAND =~                                                              
##     Im17              1.000                               1.204    0.969
##     Im18              0.994    0.041   24.143    0.000    1.197    0.856
##   FRENCH =~                                                             
##     Im6               1.000                               0.975    0.813
##     Im7               1.184    0.071   16.770    0.000    1.155    0.955
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO ~~                                                               
##     FOOD              0.418    0.050    8.393    0.000    0.416    0.416
##     ATMOS             0.730    0.082    8.912    0.000    0.467    0.467
##     PRODQUAL          0.409    0.051    8.040    0.000    0.471    0.471
##     CHOICE            0.711    0.079    9.032    0.000    0.441    0.441
##     PROF              0.743    0.071   10.465    0.000    0.653    0.653
##     BRAND             0.770    0.076   10.140    0.000    0.517    0.517
##     FRENCH            0.402    0.063    6.350    0.000    0.334    0.334
##   FOOD ~~                                                               
##     ATMOS             0.303    0.051    5.948    0.000    0.295    0.295
##     PRODQUAL          0.258    0.034    7.662    0.000    0.452    0.452
##     CHOICE            0.328    0.050    6.584    0.000    0.309    0.309
##     PROF              0.372    0.043    8.589    0.000    0.498    0.498
##     BRAND             0.318    0.047    6.801    0.000    0.325    0.325
##     FRENCH            0.463    0.047    9.829    0.000    0.585    0.585
##   ATMOS ~~                                                              
##     PRODQUAL          0.372    0.053    7.011    0.000    0.418    0.418
##     CHOICE            0.739    0.085    8.728    0.000    0.448    0.448
##     PROF              0.557    0.069    8.089    0.000    0.478    0.478
##     BRAND             0.787    0.081    9.715    0.000    0.516    0.516
##     FRENCH            0.410    0.065    6.352    0.000    0.333    0.333
##   PRODQUAL ~~                                                           
##     CHOICE            0.439    0.054    8.161    0.000    0.478    0.478
##     PROF              0.343    0.043    7.946    0.000    0.529    0.529
##     BRAND             0.479    0.053    9.046    0.000    0.566    0.566
##     FRENCH            0.210    0.037    5.622    0.000    0.306    0.306
##   CHOICE ~~                                                             
##     PROF              0.717    0.072    9.956    0.000    0.597    0.597
##     BRAND             0.817    0.079   10.362    0.000    0.519    0.519
##     FRENCH            0.286    0.060    4.735    0.000    0.225    0.225
##   PROF ~~                                                               
##     BRAND             0.667    0.066   10.040    0.000    0.601    0.601
##     FRENCH            0.328    0.051    6.438    0.000    0.366    0.366
##   BRAND ~~                                                              
##     FRENCH            0.378    0.061    6.175    0.000    0.322    0.322
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               4.995    0.056   88.560    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.983    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.844    0.000    5.035    3.787
##    .Im10              6.100    0.037  162.789    0.000    6.100    6.937
##    .Im14              6.138    0.037  165.861    0.000    6.138    7.093
##    .Im20              4.672    0.064   73.177    0.000    4.672    3.123
##    .Im21              5.139    0.058   87.970    0.000    5.139    3.751
##    .Im22              4.279    0.065   65.401    0.000    4.279    2.799
##    .Im11              5.653    0.049  115.271    0.000    5.653    4.943
##    .Im12              5.666    0.049  116.089    0.000    5.666    4.983
##    .Im13              5.448    0.052  105.615    0.000    5.448    4.524
##    .Im1               4.790    0.057   84.202    0.000    4.790    3.597
##    .Im2               4.857    0.055   88.354    0.000    4.857    3.779
##    .Im16              5.135    0.052   99.147    0.000    5.135    4.269
##    .Im19              5.145    0.048  106.948    0.000    5.145    4.574
##    .Im17              5.025    0.053   94.519    0.000    5.025    4.041
##    .Im18              4.595    0.060   76.447    0.000    4.595    3.287
##    .Im6               5.827    0.051  113.784    0.000    5.827    4.858
##    .Im7               5.753    0.052  110.826    0.000    5.753    4.756
##     DECO              0.000                               0.000    0.000
##     FOOD              0.000                               0.000    0.000
##     ATMOS             0.000                               0.000    0.000
##     PRODQUAL          0.000                               0.000    0.000
##     CHOICE            0.000                               0.000    0.000
##     PROF              0.000                               0.000    0.000
##     BRAND             0.000                               0.000    0.000
##     FRENCH            0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               0.213    0.024    8.755    0.000    0.213    0.122
##    .Im4               0.109    0.024    4.532    0.000    0.109    0.060
##    .Im5               0.747    0.049   15.217    0.000    0.747    0.422
##    .Im10              0.114    0.019    5.961    0.000    0.114    0.148
##    .Im14              0.070    0.019    3.680    0.000    0.070    0.093
##    .Im20              0.638    0.061   10.451    0.000    0.638    0.285
##    .Im21              0.725    0.057   12.672    0.000    0.725    0.386
##    .Im22              0.541    0.063    8.539    0.000    0.541    0.231
##    .Im11              0.814    0.055   14.802    0.000    0.814    0.622
##    .Im12              0.310    0.040    7.845    0.000    0.310    0.240
##    .Im13              0.390    0.045    8.765    0.000    0.390    0.269
##    .Im1               0.070    0.050    1.394    0.163    0.070    0.040
##    .Im2               0.317    0.044    7.233    0.000    0.317    0.192
##    .Im16              0.599    0.052   11.498    0.000    0.599    0.414
##    .Im19              0.338    0.045    7.457    0.000    0.338    0.267
##    .Im17              0.095    0.045    2.112    0.035    0.095    0.062
##    .Im18              0.521    0.055    9.540    0.000    0.521    0.267
##    .Im6               0.487    0.056    8.677    0.000    0.487    0.339
##    .Im7               0.128    0.067    1.930    0.054    0.128    0.088
##     DECO              1.528    0.107   14.326    0.000    1.000    1.000
##     FOOD              0.659    0.049   13.328    0.000    1.000    1.000
##     ATMOS             1.599    0.138   11.623    0.000    1.000    1.000
##     PRODQUAL          0.494    0.067    7.361    0.000    1.000    1.000
##     CHOICE            1.704    0.118   14.388    0.000    1.000    1.000
##     PROF              0.849    0.088    9.638    0.000    1.000    1.000
##     BRAND             1.451    0.104   13.988    0.000    1.000    1.000
##     FRENCH            0.952    0.095   10.058    0.000    1.000    1.000

2.3.2.1 Discussion of global fit measures

Chi square: p-value > 0.05

RMSEA RMSEA <= 0.05 Good fit 0.05 < RMSEA <= 0.08 Acceptable fit 0.08 < RMSEA <= 0.10 Bad fit RMSEA > 0.1 Unacceptable fit

CFI CFI < 0.90 definitely reject model 0.90 < CFI < 0.95 high underrejection rates for misspecified models CFI > 0.95 accept model

2.3.2.2 local fit measures

# semPaths(CFA_fit_img_7f, what = "path", whatLabels = "std", style = "mx",
#          rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
#          nCharNodes = 7,shapeMan = "rectangle", sizeMan = 8, sizeMan2 = 5, 
#          curvePivot=TRUE, edge.label.cex = 1.2, edge.color = "skyblue4")


semPaths(CFA_fit_img_7f, what = "path", whatLabels = "std", style = "mx",
         rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
         nCharNodes = 7,shapeMan = "rectangle", 
         sizeMan = 4, sizeMan2 = 3, sizeInt = 2, sizeLat = 6, asize = 1.5,
         curvePivot=TRUE, edge.label.cex = .8, edge.color = "skyblue4"
         )

lambda = inspect(CFA_fit_img_7f, what="std")$lambda
theta = inspect(CFA_fit_img_7f, what="std")$theta

# create lambda matrix with ones instead of std.all
ones <- lambda
ones[ones>0] <- 1

# a matrix with dimensions of lambda matrix but with lambdas replaced by thetas
theta_lb <- theta %*% ones
2.3.2.2.1 Indicator reliability criterion
# calculate indicator reliabilities (should be larger than 0.4)
indicrel <- lambda^2/(lambda^2 + theta_lb)
# indicrel

# replace all values satisfying condition with NaN for visibility
indicrel_fail <- indicrel
indicrel_fail[indicrel_fail>.4] <- NaN
indicrel_fail
##      DECO FOOD ATMOS PRODQU CHOICE PROF BRAND FRENCH
## Im3   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im4   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im5   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im10  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im14  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im20  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im21  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im22  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im11  NaN  NaN   NaN  0.378    NaN  NaN   NaN    NaN
## Im12  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im13  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im1   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im2   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im16  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im19  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im17  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im18  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im6   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
## Im7   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN
2.3.2.2.2 Construct reliability criterion
# calculate construct reliability (should be above .6)
constrrel <- (t(lambda) %*% ones)^2 / ((t(lambda) %*% ones)^2 + t(theta_lb) %*% ones )
# constrrel

# replace all values satisfying condition with NaN for visibility
constrrel_fail <- constrrel
constrrel_fail[constrrel_fail>.6] <- NaN
constrrel_fail
##          DECO FOOD ATMOS PRODQUAL CHOICE PROF BRAND FRENCH
## DECO      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## FOOD      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## ATMOS     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## PRODQUAL  NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## CHOICE    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## PROF      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## BRAND     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## FRENCH    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
2.3.2.2.3 Average Variance Extracted criterion
# calculate Average Variance Extracted (should be above .5)
AVE <- (t(lambda) %*% lambda) / (t(lambda) %*% lambda + t(theta_lb) %*% ones )
# avgvar

# replace all values satisfying condition with NaN for visibility
AVE_fail <- AVE
AVE_fail[AVE_fail>.5] <- NaN
AVE_fail
##          DECO FOOD ATMOS PRODQUAL CHOICE PROF BRAND FRENCH
## DECO      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## FOOD      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## ATMOS     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## PRODQUAL  NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## CHOICE    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## PROF      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## BRAND     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
## FRENCH    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN
2.3.2.2.4 Fornell-Larcker Criteria
# correlations between constructs (factors...) should be lower than .7
psi = inspect(CFA_fit_img_7f, what="std")$psi
psi_fail <- psi
psi_fail[psi_fail<.7] <- NaN
psi_fail
##          DECO FOOD ATMOS PRODQU CHOICE PROF BRAND FRENCH
## DECO        1                                           
## FOOD      NaN    1                                      
## ATMOS     NaN  NaN     1                                
## PRODQUAL  NaN  NaN   NaN      1                         
## CHOICE    NaN  NaN   NaN    NaN      1                  
## PROF      NaN  NaN   NaN    NaN    NaN    1             
## BRAND     NaN  NaN   NaN    NaN    NaN  NaN     1       
## FRENCH    NaN  NaN   NaN    NaN    NaN  NaN   NaN      1
# AVE should be higher than squared correlations between constructs

# replace diagonal of psi matrix with AVE values
psi2 <- psi - psi * diag(1,nrow(psi),ncol(psi)) + diag(AVE) * diag(1,nrow(AVE),ncol(AVE))

# create matrix with columns filled with AVE
AVE_full <- AVE
AVE_full[is.na(AVE_full)] <- 0 #replace NAs with 0s
AVE_full <- AVE_full^0 %*% AVE_full # multiply a matrix full of ones with AVE_full to get columns filled with AVE

# substract matrices, replace all values satisfying positive condition (AVE > psi) with NaN
AVEpsi_fail <- AVE_full - psi2
# AVE_full - psi2
AVEpsi_fail[AVEpsi_fail >= 0] <- NaN

AVE_full
##               DECO      FOOD     ATMOS PRODQUAL    CHOICE      PROF     BRAND
## DECO     0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## FOOD     0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## ATMOS    0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## PRODQUAL 0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## CHOICE   0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## PROF     0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## BRAND    0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
## FRENCH   0.7983933 0.8796192 0.6990218 0.622977 0.8842421 0.6598497 0.8358722
##            FRENCH
## DECO     0.786707
## FOOD     0.786707
## ATMOS    0.786707
## PRODQUAL 0.786707
## CHOICE   0.786707
## PROF     0.786707
## BRAND    0.786707
## FRENCH   0.786707
psi
##           DECO  FOOD ATMOS PRODQU CHOICE  PROF BRAND FRENCH
## DECO     1.000                                             
## FOOD     0.416 1.000                                       
## ATMOS    0.467 0.295 1.000                                 
## PRODQUAL 0.471 0.452 0.418  1.000                          
## CHOICE   0.441 0.309 0.448  0.478  1.000                   
## PROF     0.653 0.498 0.478  0.529  0.597 1.000             
## BRAND    0.517 0.325 0.516  0.566  0.519 0.601 1.000       
## FRENCH   0.334 0.585 0.333  0.306  0.225 0.366 0.322  1.000
AVEpsi_fail
##            DECO   FOOD  ATMOS PRODQU CHOICE   PROF  BRAND FRENCH
## DECO          .                                                 
## FOOD        NaN      .                                          
## ATMOS       NaN    NaN      .                                   
## PRODQUAL    NaN    NaN    NaN      .                            
## CHOICE      NaN    NaN    NaN    NaN      .                     
## PROF        NaN    NaN    NaN    NaN    NaN      .              
## BRAND       NaN    NaN    NaN    NaN    NaN    NaN      .       
## FRENCH      NaN    NaN    NaN    NaN    NaN    NaN    NaN      .

2.3.2.3 Modification indices

arrange(modificationindices(CFA_fit_img_7f),-mi)
##          lhs op  rhs     mi    epc sepc.lv sepc.all sepc.nox
## 1      BRAND =~ Im13 23.832  0.220   0.265    0.220    0.220
## 2       Im11 ~~ Im13 21.323 -0.191  -0.191   -0.338   -0.338
## 3      BRAND =~ Im12 17.245 -0.179  -0.216   -0.190   -0.190
## 4       Im21 ~~ Im22 15.139 -0.285  -0.285   -0.455   -0.455
## 5     CHOICE =~ Im20 14.777 -0.151  -0.197   -0.132   -0.132
## 6     CHOICE =~ Im13 13.970  0.133   0.174    0.144    0.144
## 7       Im11 ~~ Im12 13.307  0.145   0.145    0.288    0.288
## 8       FOOD =~ Im11 12.742  0.215   0.174    0.152    0.152
## 9       Im20 ~~ Im21 11.455  0.228   0.228    0.335    0.335
## 10     ATMOS =~ Im12 10.952 -0.115  -0.145   -0.127   -0.127
## 11      Im13 ~~  Im1 10.707  0.068   0.068    0.409    0.409
## 12    CHOICE =~ Im12 10.663 -0.111  -0.145   -0.127   -0.127
## 13      Im13 ~~ Im17  9.392  0.068   0.068    0.355    0.355
## 14       Im4 ~~ Im17  9.096 -0.046  -0.046   -0.446   -0.446
## 15     BRAND =~ Im20  8.230 -0.133  -0.160   -0.107   -0.107
## 16      Im10 ~~ Im16  7.956  0.046   0.046    0.175    0.175
## 17     BRAND =~ Im22  7.656  0.131   0.158    0.103    0.103
## 18    FRENCH =~ Im22  7.278  0.136   0.132    0.087    0.087
## 19     ATMOS =~ Im11  6.691  0.101   0.128    0.112    0.112
## 20      Im22 ~~ Im12  6.644 -0.076  -0.076   -0.185   -0.185
## 21    CHOICE =~  Im5  6.302  0.086   0.112    0.084    0.084
## 22     BRAND =~  Im4  6.247 -0.067  -0.081   -0.060   -0.060
## 23  PRODQUAL =~  Im5  6.120  0.171   0.120    0.091    0.091
## 24      Im14 ~~ Im16  6.048 -0.039  -0.039   -0.191   -0.191
## 25      Im10 ~~  Im6  6.011 -0.035  -0.035   -0.147   -0.147
## 26     BRAND =~  Im5  5.755  0.095   0.115    0.086    0.086
## 27       Im3 ~~  Im1  5.614 -0.034  -0.034   -0.281   -0.281
## 28      DECO =~  Im7  5.531 -0.093  -0.114   -0.095   -0.095
## 29      DECO =~  Im6  5.531  0.078   0.097    0.080    0.080
## 30     ATMOS =~  Im5  5.517  0.089   0.113    0.085    0.085
## 31      DECO =~ Im18  5.361  0.099   0.123    0.088    0.088
## 32      DECO =~ Im17  5.361 -0.100  -0.124   -0.099   -0.099
## 33    FRENCH =~ Im20  5.231 -0.113  -0.111   -0.074   -0.074
## 34      Im22 ~~ Im11  5.182  0.087   0.087    0.131    0.131
## 35      FOOD =~ Im13  5.137 -0.127  -0.103   -0.086   -0.086
## 36       Im3 ~~  Im4  5.131  0.162   0.162    1.061    1.061
## 37      DECO =~ Im20  5.124 -0.096  -0.118   -0.079   -0.079
## 38     BRAND =~  Im7  5.064 -0.090  -0.108   -0.089   -0.089
## 39     BRAND =~  Im6  5.064  0.076   0.091    0.076    0.076
## 40      Im22 ~~  Im1  5.051  0.056   0.056    0.290    0.290
## 41      Im13 ~~ Im16  5.040 -0.067  -0.067   -0.138   -0.138
## 42      Im11 ~~  Im6  4.965 -0.069  -0.069   -0.109   -0.109
## 43      Im10 ~~ Im13  4.793 -0.031  -0.031   -0.146   -0.146
## 44       Im4 ~~ Im18  4.776  0.040   0.040    0.166    0.166
## 45       Im5 ~~  Im6  4.758 -0.065  -0.065   -0.108   -0.108
## 46       Im3 ~~ Im22  4.749  0.049   0.049    0.143    0.143
## 47      Im10 ~~  Im7  4.706  0.032   0.032    0.265    0.265
## 48    CHOICE =~ Im22  4.697  0.087   0.113    0.074    0.074
## 49       Im3 ~~  Im5  4.674 -0.071  -0.071   -0.178   -0.178
## 50      Im13 ~~  Im2  4.562 -0.044  -0.044   -0.125   -0.125
## 51      FOOD =~  Im5  4.520  0.117   0.095    0.072    0.072
## 52      Im20 ~~ Im17  4.488 -0.057  -0.057   -0.230   -0.230
## 53     ATMOS =~  Im4  4.309 -0.052  -0.066   -0.049   -0.049
## 54      DECO =~ Im22  4.302  0.090   0.111    0.072    0.072
## 55      PROF =~  Im2  4.223  0.170   0.157    0.122    0.122
## 56      PROF =~  Im1  4.223 -0.192  -0.177   -0.133   -0.133
## 57      Im20 ~~  Im6  4.215 -0.064  -0.064   -0.115   -0.115
## 58  PRODQUAL =~  Im1  4.027  0.145   0.102    0.077    0.077
## 59  PRODQUAL =~  Im2  4.027 -0.129  -0.090   -0.070   -0.070
## 60      PROF =~ Im12  3.965 -0.117  -0.108   -0.095   -0.095
## 61      Im14 ~~  Im6  3.823  0.027   0.027    0.149    0.149
## 62      FOOD =~  Im6  3.804 -0.271  -0.220   -0.183   -0.183
## 63      FOOD =~  Im7  3.804  0.321   0.260    0.215    0.215
## 64       Im2 ~~ Im17  3.691  0.033   0.033    0.192    0.192
## 65       Im4 ~~  Im6  3.588  0.033   0.033    0.143    0.143
## 66       Im3 ~~ Im17  3.342  0.028   0.028    0.198    0.198
## 67      Im11 ~~  Im1  3.335 -0.045  -0.045   -0.189   -0.189
## 68       Im5 ~~  Im1  3.275  0.043   0.043    0.188    0.188
## 69      PROF =~ Im20  3.209 -0.113  -0.104   -0.069   -0.069
## 70    CHOICE =~ Im21  3.202  0.067   0.087    0.064    0.064
## 71       Im3 ~~  Im2  3.151  0.026   0.026    0.100    0.100
## 72      Im20 ~~ Im13  3.089  0.055   0.055    0.111    0.111
## 73      Im18 ~~  Im6  3.067  0.046   0.046    0.091    0.091
## 74      Im14 ~~  Im7  3.014 -0.026  -0.026   -0.272   -0.272
## 75      Im11 ~~ Im17  2.983 -0.045  -0.045   -0.163   -0.163
## 76    FRENCH =~ Im11  2.967  0.080   0.078    0.069    0.069
## 77     ATMOS =~ Im13  2.904  0.062   0.078    0.065    0.065
## 78       Im5 ~~  Im7  2.807  0.048   0.048    0.154    0.154
## 79      Im10 ~~ Im11  2.784  0.028   0.028    0.093    0.093
## 80      Im20 ~~  Im1  2.774 -0.042  -0.042   -0.198   -0.198
## 81       Im5 ~~ Im14  2.721  0.026   0.026    0.115    0.115
## 82       Im1 ~~ Im17  2.656 -0.029  -0.029   -0.359   -0.359
## 83    FRENCH =~ Im19  2.578  0.085   0.083    0.074    0.074
## 84    FRENCH =~ Im16  2.578 -0.082  -0.080   -0.066   -0.066
## 85       Im2 ~~ Im16  2.577  0.038   0.038    0.088    0.088
## 86    CHOICE =~ Im14  2.577  0.027   0.035    0.041    0.041
## 87    CHOICE =~ Im10  2.577 -0.026  -0.035   -0.039   -0.039
## 88      Im21 ~~ Im18  2.573 -0.052  -0.052   -0.084   -0.084
## 89       Im4 ~~ Im11  2.457 -0.034  -0.034   -0.113   -0.113
## 90      PROF =~ Im13  2.455  0.096   0.089    0.074    0.074
## 91      Im22 ~~ Im19  2.419 -0.049  -0.049   -0.115   -0.115
## 92       Im4 ~~ Im22  2.418 -0.034  -0.034   -0.138   -0.138
## 93       Im3 ~~ Im20  2.350 -0.034  -0.034   -0.093   -0.093
## 94      Im12 ~~  Im7  2.333  0.036   0.036    0.178    0.178
## 95      DECO =~ Im12  2.244 -0.054  -0.066   -0.058   -0.058
## 96      Im11 ~~  Im7  2.221  0.044   0.044    0.137    0.137
## 97      Im19 ~~ Im17  2.187  0.035   0.035    0.193    0.193
## 98  PRODQUAL =~ Im16  2.157 -0.133  -0.094   -0.078   -0.078
## 99  PRODQUAL =~ Im19  2.157  0.140   0.098    0.087    0.087
## 100     Im10 ~~ Im17  2.035 -0.017  -0.017   -0.161   -0.161
## 101     Im14 ~~ Im17  1.988  0.016   0.016    0.201    0.201
## 102      Im4 ~~  Im1  1.978  0.020   0.020    0.227    0.227
## 103      Im3 ~~ Im12  1.906 -0.024  -0.024   -0.092   -0.092
## 104     Im12 ~~  Im6  1.902 -0.033  -0.033   -0.084   -0.084
## 105   CHOICE =~ Im16  1.817  0.068   0.089    0.074    0.074
## 106   CHOICE =~ Im19  1.817 -0.071  -0.093   -0.083   -0.083
## 107     Im14 ~~  Im2  1.800  0.015   0.015    0.098    0.098
## 108     Im18 ~~  Im7  1.743 -0.033  -0.033   -0.128   -0.128
## 109   FRENCH =~  Im5  1.735  0.059   0.057    0.043    0.043
## 110     Im11 ~~  Im2  1.713  0.033   0.033    0.064    0.064
## 111    ATMOS =~ Im17  1.684 -0.057  -0.073   -0.058   -0.058
## 112    ATMOS =~ Im18  1.684  0.057   0.072    0.052    0.052
## 113     Im19 ~~ Im18  1.680 -0.034  -0.034   -0.082   -0.082
## 114    BRAND =~  Im3  1.663  0.034   0.041    0.031    0.031
## 115      Im3 ~~ Im14  1.662 -0.012  -0.012   -0.102   -0.102
## 116   FRENCH =~  Im2  1.647  0.039   0.038    0.030    0.030
## 117   FRENCH =~  Im1  1.647 -0.045  -0.043   -0.033   -0.033
## 118     Im22 ~~  Im2  1.646 -0.032  -0.032   -0.077   -0.077
## 119     Im22 ~~ Im18  1.613  0.041   0.041    0.077    0.077
## 120     FOOD =~  Im2  1.604  0.050   0.040    0.031    0.031
## 121     FOOD =~  Im1  1.604 -0.056  -0.046   -0.034   -0.034
## 122 PRODQUAL =~  Im4  1.602 -0.058  -0.041   -0.030   -0.030
## 123     Im22 ~~ Im13  1.588 -0.040  -0.040   -0.086   -0.086
## 124      Im4 ~~  Im2  1.557 -0.018  -0.018   -0.095   -0.095
## 125     Im14 ~~ Im21  1.552  0.021   0.021    0.091    0.091
## 126      Im5 ~~ Im16  1.529 -0.043  -0.043   -0.064   -0.064
## 127     Im12 ~~ Im17  1.526 -0.026  -0.026   -0.151   -0.151
## 128     Im10 ~~ Im18  1.519  0.018   0.018    0.072    0.072
## 129 PRODQUAL =~ Im22  1.519 -0.095  -0.067   -0.044   -0.044
## 130      Im2 ~~ Im18  1.481 -0.026  -0.026   -0.063   -0.063
## 131     Im12 ~~ Im13  1.475  0.089   0.089    0.257    0.257
## 132 PRODQUAL =~  Im6  1.473 -0.073  -0.052   -0.043   -0.043
## 133 PRODQUAL =~  Im7  1.473  0.087   0.061    0.051    0.051
## 134     Im14 ~~ Im18  1.464 -0.017  -0.017   -0.088   -0.088
## 135     PROF =~ Im22  1.445  0.077   0.071    0.047    0.047
## 136     DECO =~ Im13  1.431  0.045   0.055    0.046    0.046
## 137 PRODQUAL =~ Im17  1.422  0.109   0.076    0.062    0.062
## 138 PRODQUAL =~ Im18  1.422 -0.108  -0.076   -0.054   -0.054
## 139      Im4 ~~ Im12  1.373  0.019   0.019    0.106    0.106
## 140    ATMOS =~  Im1  1.355  0.043   0.054    0.040    0.040
## 141    ATMOS =~  Im2  1.355 -0.038  -0.048   -0.037   -0.037
## 142     Im22 ~~  Im7  1.342  0.035   0.035    0.133    0.133
## 143      Im3 ~~ Im18  1.316 -0.022  -0.022   -0.065   -0.065
## 144      Im3 ~~ Im11  1.290  0.025   0.025    0.061    0.061
## 145     PROF =~  Im4  1.267 -0.055  -0.050   -0.037   -0.037
## 146     Im13 ~~ Im19  1.244  0.029   0.029    0.080    0.080
## 147      Im3 ~~ Im10  1.231  0.011   0.011    0.071    0.071
## 148     Im10 ~~ Im12  1.205  0.014   0.014    0.077    0.077
## 149     Im12 ~~ Im16  1.199  0.030   0.030    0.071    0.071
## 150    BRAND =~ Im11  1.070 -0.047  -0.056   -0.049   -0.049
## 151    BRAND =~  Im2  1.067  0.042   0.051    0.039    0.039
## 152    BRAND =~  Im1  1.067 -0.048  -0.057   -0.043   -0.043
## 153     Im14 ~~ Im12  1.061 -0.013  -0.013   -0.090   -0.090
## 154      Im4 ~~  Im7  1.054 -0.017  -0.017   -0.147   -0.147
## 155 PRODQUAL =~ Im20  1.037  0.077   0.054    0.036    0.036
## 156     Im21 ~~ Im17  1.017  0.027   0.027    0.101    0.101
## 157     Im20 ~~ Im19  0.988  0.031   0.031    0.068    0.068
## 158     PROF =~ Im17  0.986  0.084   0.077    0.062    0.062
## 159     PROF =~ Im18  0.986 -0.083  -0.077   -0.055   -0.055
## 160     Im19 ~~  Im7  0.949  0.025   0.025    0.122    0.122
## 161     PROF =~  Im7  0.940 -0.060  -0.055   -0.045   -0.045
## 162     PROF =~  Im6  0.940  0.050   0.046    0.039    0.039
## 163     Im13 ~~ Im18  0.925 -0.026  -0.026   -0.057   -0.057
## 164      Im5 ~~  Im2  0.918 -0.023  -0.023   -0.047   -0.047
## 165     Im14 ~~ Im13  0.889  0.013   0.013    0.079    0.079
## 166     Im16 ~~  Im7  0.880 -0.027  -0.027   -0.097   -0.097
## 167      Im1 ~~ Im18  0.847  0.019   0.019    0.101    0.101
## 168    ATMOS =~  Im3  0.826  0.022   0.028    0.022    0.022
## 169     FOOD =~ Im20  0.814 -0.052  -0.042   -0.028   -0.028
## 170     Im22 ~~  Im6  0.806  0.028   0.028    0.054    0.054
## 171      Im5 ~~ Im19  0.786 -0.026  -0.026   -0.052   -0.052
## 172    BRAND =~ Im16  0.775 -0.050  -0.060   -0.050   -0.050
## 173    BRAND =~ Im19  0.775  0.052   0.063    0.056    0.056
## 174      Im3 ~~ Im19  0.774  0.016   0.016    0.061    0.061
## 175    BRAND =~ Im14  0.773  0.017   0.020    0.023    0.023
## 176    BRAND =~ Im10  0.773 -0.016  -0.020   -0.022   -0.022
## 177     Im10 ~~ Im19  0.726 -0.012  -0.012   -0.062   -0.062
## 178   FRENCH =~ Im13  0.706 -0.035  -0.034   -0.028   -0.028
## 179     PROF =~  Im5  0.688  0.055   0.051    0.038    0.038
## 180      Im4 ~~  Im5  0.687  0.029   0.029    0.103    0.103
## 181     Im20 ~~ Im12  0.675  0.024   0.024    0.054    0.054
## 182      Im2 ~~  Im7  0.669  0.016   0.016    0.079    0.079
## 183      Im3 ~~  Im7  0.637 -0.014  -0.014   -0.084   -0.084
## 184 PRODQUAL =~ Im14  0.619  0.031   0.022    0.025    0.025
## 185 PRODQUAL =~ Im10  0.619 -0.030  -0.021   -0.024   -0.024
## 186     Im21 ~~  Im7  0.611 -0.023  -0.023   -0.076   -0.076
## 187      Im5 ~~ Im22  0.596  0.028   0.028    0.044    0.044
## 188      Im4 ~~ Im10  0.595 -0.007  -0.007   -0.066   -0.066
## 189     Im20 ~~ Im18  0.582  0.025   0.025    0.043    0.043
## 190     Im10 ~~  Im2  0.581 -0.008  -0.008   -0.045   -0.045
## 191      Im4 ~~ Im16  0.579  0.016   0.016    0.062    0.062
## 192     Im21 ~~  Im2  0.574  0.019   0.019    0.039    0.039
## 193     Im10 ~~ Im21  0.544 -0.013  -0.013   -0.043   -0.043
## 194     Im14 ~~ Im22  0.525 -0.012  -0.012   -0.062   -0.062
## 195     Im13 ~~  Im7  0.518 -0.018  -0.018   -0.080   -0.080
## 196      Im5 ~~ Im11  0.501  0.026   0.026    0.033    0.033
## 197     Im20 ~~ Im11  0.498  0.027   0.027    0.038    0.038
## 198     Im12 ~~  Im2  0.492 -0.013  -0.013   -0.043   -0.043
## 199     Im11 ~~ Im18  0.492  0.023   0.023    0.035    0.035
## 200   CHOICE =~  Im3  0.483 -0.015  -0.020   -0.015   -0.015
## 201      Im1 ~~ Im16  0.483 -0.017  -0.017   -0.082   -0.082
## 202     Im21 ~~ Im12  0.479  0.020   0.020    0.043    0.043
## 203      Im5 ~~ Im10  0.474 -0.011  -0.011   -0.039   -0.039
## 204   CHOICE =~ Im11  0.473 -0.026  -0.034   -0.030   -0.030
## 205     Im10 ~~ Im20  0.466  0.012   0.012    0.043    0.043
## 206     PROF =~  Im3  0.453  0.032   0.029    0.022    0.022
## 207   FRENCH =~  Im3  0.431 -0.019  -0.018   -0.014   -0.014
## 208     FOOD =~  Im3  0.430 -0.023  -0.019   -0.014   -0.014
## 209     FOOD =~ Im19  0.429  0.047   0.038    0.034    0.034
## 210     FOOD =~ Im16  0.429 -0.045  -0.037   -0.031   -0.031
## 211     PROF =~ Im11  0.410  0.039   0.036    0.032    0.032
## 212      Im5 ~~ Im20  0.382  0.023   0.023    0.033    0.033
## 213     Im13 ~~  Im6  0.377  0.016   0.016    0.036    0.036
## 214     Im14 ~~ Im19  0.372  0.009   0.009    0.056    0.056
## 215     Im21 ~~ Im13  0.372 -0.019  -0.019   -0.036   -0.036
## 216     PROF =~ Im21  0.371  0.036   0.033    0.024    0.024
## 217   CHOICE =~  Im4  0.369 -0.014  -0.018   -0.013   -0.013
## 218     Im22 ~~ Im17  0.360  0.016   0.016    0.071    0.071
## 219   FRENCH =~ Im21  0.360 -0.029  -0.028   -0.020   -0.020
## 220     Im21 ~~ Im11  0.346 -0.022  -0.022   -0.029   -0.029
## 221      Im5 ~~ Im17  0.345  0.015   0.015    0.055    0.055
## 222     Im14 ~~ Im20  0.330 -0.010  -0.010   -0.045   -0.045
## 223     FOOD =~ Im21  0.320  0.032   0.026    0.019    0.019
## 224   CHOICE =~ Im17  0.309  0.023   0.030    0.024    0.024
## 225   CHOICE =~ Im18  0.309 -0.023  -0.030   -0.021   -0.021
## 226      Im3 ~~ Im13  0.299  0.010   0.010    0.035    0.035
## 227      Im4 ~~ Im19  0.281 -0.010  -0.010   -0.050   -0.050
## 228     Im22 ~~ Im16  0.277  0.019   0.019    0.033    0.033
## 229      Im5 ~~ Im21  0.277 -0.019  -0.019   -0.026   -0.026
## 230      Im4 ~~ Im14  0.246  0.005   0.005    0.053    0.053
## 231     DECO =~ Im11  0.238  0.019   0.024    0.021    0.021
## 232     Im16 ~~ Im18  0.231 -0.015  -0.015   -0.026   -0.026
## 233     Im17 ~~  Im7  0.228 -0.010  -0.010   -0.094   -0.094
## 234     Im21 ~~ Im16  0.227 -0.017  -0.017   -0.026   -0.026
## 235     Im20 ~~ Im22  0.211  0.043   0.043    0.074    0.074
## 236     Im17 ~~  Im6  0.208  0.010   0.010    0.045    0.045
## 237     Im20 ~~  Im2  0.203 -0.011  -0.011   -0.025   -0.025
## 238      Im2 ~~ Im19  0.202 -0.009  -0.009   -0.029   -0.029
## 239     FOOD =~  Im4  0.194 -0.016  -0.013   -0.010   -0.010
## 240      Im4 ~~ Im13  0.168 -0.007  -0.007   -0.035   -0.035
## 241     Im14 ~~ Im11  0.165  0.007   0.007    0.028    0.028
## 242     FOOD =~ Im22  0.151  0.023   0.018    0.012    0.012
## 243     Im12 ~~ Im18  0.146 -0.010  -0.010   -0.024   -0.024
## 244     Im20 ~~ Im16  0.145  0.014   0.014    0.022    0.022
## 245     Im21 ~~  Im1  0.132  0.009   0.009    0.040    0.040
## 246    ATMOS =~ Im19  0.131 -0.017  -0.022   -0.019   -0.019
## 247    ATMOS =~ Im16  0.131  0.016   0.021    0.017    0.017
## 248      Im4 ~~ Im20  0.129  0.008   0.008    0.029    0.029
## 249      Im1 ~~  Im7  0.126 -0.007  -0.007   -0.075   -0.075
## 250     Im12 ~~  Im1  0.110 -0.006  -0.006   -0.044   -0.044
## 251      Im3 ~~  Im6  0.106  0.006   0.006    0.018    0.018
## 252      Im2 ~~  Im6  0.094 -0.006  -0.006   -0.016   -0.016
## 253   FRENCH =~ Im18  0.094  0.013   0.012    0.009    0.009
## 254   FRENCH =~ Im17  0.094 -0.013  -0.013   -0.010   -0.010
## 255     Im10 ~~  Im1  0.093 -0.003  -0.003   -0.038   -0.038
## 256 PRODQUAL =~ Im21  0.088  0.021   0.015    0.011    0.011
## 257      Im5 ~~ Im18  0.083  0.009   0.009    0.014    0.014
## 258     Im20 ~~  Im7  0.082  0.009   0.009    0.030    0.030
## 259     DECO =~  Im2  0.075  0.009   0.011    0.008    0.008
## 260     DECO =~  Im1  0.075 -0.010  -0.012   -0.009   -0.009
## 261      Im5 ~~ Im12  0.067  0.007   0.007    0.015    0.015
## 262     Im14 ~~  Im1  0.063 -0.003  -0.003   -0.039   -0.039
## 263    ATMOS =~ Im14  0.061  0.004   0.006    0.006    0.006
## 264    ATMOS =~ Im10  0.061 -0.004  -0.006   -0.006   -0.006
## 265     Im21 ~~ Im19  0.060  0.008   0.008    0.015    0.015
## 266      Im4 ~~ Im21  0.059  0.005   0.005    0.018    0.018
## 267     PROF =~ Im10  0.058  0.008   0.007    0.008    0.008
## 268     PROF =~ Im14  0.058 -0.008  -0.007   -0.009   -0.009
## 269      Im1 ~~ Im19  0.056 -0.005  -0.005   -0.034   -0.034
## 270   CHOICE =~  Im6  0.055 -0.007  -0.009   -0.007   -0.007
## 271   CHOICE =~  Im7  0.055  0.008   0.010    0.008    0.008
## 272     DECO =~ Im14  0.052  0.005   0.006    0.007    0.007
## 273     DECO =~ Im10  0.052 -0.005  -0.006   -0.006   -0.006
## 274     Im10 ~~ Im22  0.051  0.004   0.004    0.015    0.015
## 275   FRENCH =~ Im12  0.045 -0.008  -0.008   -0.007   -0.007
## 276      Im1 ~~  Im6  0.039 -0.004  -0.004   -0.021   -0.021
## 277    ATMOS =~  Im6  0.036 -0.006  -0.008   -0.007   -0.007
## 278    ATMOS =~  Im7  0.036  0.008   0.010    0.008    0.008
## 279     Im12 ~~ Im19  0.023 -0.004  -0.004   -0.011   -0.011
## 280     DECO =~ Im21  0.016  0.005   0.006    0.005    0.005
## 281     FOOD =~ Im17  0.015  0.006   0.005    0.004    0.004
## 282     FOOD =~ Im18  0.015 -0.006  -0.005   -0.004   -0.004
## 283      Im3 ~~ Im21  0.013 -0.003  -0.003   -0.006   -0.006
## 284     Im16 ~~  Im6  0.011 -0.003  -0.003   -0.006   -0.006
## 285     Im19 ~~  Im6  0.009 -0.002  -0.002   -0.006   -0.006
## 286     Im11 ~~ Im19  0.006 -0.002  -0.002   -0.005   -0.005
## 287     Im21 ~~  Im6  0.006  0.002   0.002    0.004    0.004
## 288     DECO =~ Im16  0.004  0.004   0.005    0.004    0.004
## 289     DECO =~ Im19  0.004 -0.004  -0.005   -0.004   -0.004
## 290     Im11 ~~ Im16  0.002  0.002   0.002    0.002    0.002
## 291   FRENCH =~ Im14  0.001  0.001   0.001    0.002    0.002
## 292   FRENCH =~ Im10  0.001 -0.001  -0.001   -0.002   -0.002
## 293      Im3 ~~ Im16  0.001 -0.001  -0.001   -0.002   -0.002
## 294   FRENCH =~  Im4  0.001 -0.001  -0.001    0.000    0.000
## 295     Im16 ~~ Im17  0.000  0.000   0.000    0.002    0.002
## 296 PRODQUAL =~  Im3  0.000  0.000   0.000    0.000    0.000
## 297      Im5 ~~ Im13  0.000  0.000   0.000    0.000    0.000
## 298    BRAND =~ Im21  0.000  0.000   0.000    0.000    0.000
## 299     FOOD =~ Im12  0.000  0.000   0.000    0.000    0.000

2.4 SEM

## don't actually think we need this as we can use the full data for confirmatory and path analysis
# data_img_EFA2
# data.frame(EFA_PAFn[[3]]$scores)
# 
# numcol_data_img_EFA = dim(data_img_EFA2)[2]
# numcol_scores = dim(EFA_PAFn[[3]]$scores)[2]
# numcol_data_img_EFA
# numcol_scores
# 
# CFA_data = cbind(data_img_EFA2, EFA_PAFn[[3]]$scores, survey_excl_img2["SAT_1"])
# CFA_data
# # colnames(CFA_data)[23:29] = c("Gourmet food", "Brand image", "Choice range", "Relaxed atmosphere", "Decoration", "Product quality", "Frenchness")
# 
# # colnames(CFA_data)[numcol_data_img_EFA:(numcol_data_img_EFA + numcol_scores)] = c("FOOD", "BRAND", "CHOICE", "ATMOS", "DECO")
# colnames(CFA_data)[numcol_data_img_EFA:(numcol_data_img_EFA + numcol_scores)] = c("FOOD", "BRAND", "CHOICE", "ATMOS", "DECO", "QUAL", "FRENCH")
# 
# CFA_data

2.4.1 model

# missing Im8, Im15, Im9

model_SEM <- "
DECO =~ Im3 + Im4 + Im5
FOOD =~ Im10 + Im14
ATMOS =~ Im20 + Im21 + Im22
PRODQUAL =~ Im11 + Im12 + Im13
CHOICE =~ Im1 + Im2  
PROF =~ Im16 + Im19
BRAND =~ Im17 + Im18
FRENCH =~ Im6 + Im7

AFCOM =~ COM_A1 + COM_A2 + COM_A3 + COM_A4
SAT =~ SAT_1 + SAT_2 + SAT_3
RI =~ C_REP1 + C_REP2 + C_REP3
COI =~ C_CR1 + C_CR3 + C_CR4

SAT ~ DECO + FOOD + ATMOS + PRODQUAL + CHOICE + PROF + BRAND + FRENCH + Im8 + Im15 + Im9
AFCOM ~ DECO + FOOD + ATMOS + PRODQUAL + CHOICE + PROF + BRAND + FRENCH + Im8 + Im15 + Im9
"

2.4.2 linear regression

# # linear regression
# lm_SAT_1 <-  lm (model_SAT_1, data = survey) 
# summary(lm_SAT_1)
# # note: lm deletes all missing variables in the Xs! (not in Ys) (see help lavOptions)

2.4.3 path analysis

# path analysis
SEM_fit <- cfa(model_SEM, data=survey, missing="ML")
summary(SEM_fit, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 170 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                       164
## 
##                                                   Used       Total
##   Number of observations                           523         553
##   Number of missing patterns                       116            
## 
## Model Test User Model:
##                                                       
##   Test statistic                              1947.166
##   Degrees of freedom                               492
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             12715.671
##   Degrees of freedom                               592
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.880
##   Tucker-Lewis Index (TLI)                       0.856
##                                                       
##   Robust Comparative Fit Index (CFI)             0.882
##   Robust Tucker-Lewis Index (TLI)                0.858
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -21247.689
##   Loglikelihood unrestricted model (H1)     -20274.106
##                                                       
##   Akaike (AIC)                               42823.378
##   Bayesian (BIC)                             43521.950
##   Sample-size adjusted Bayesian (SABIC)      43001.375
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.075
##   90 Percent confidence interval - lower         0.072
##   90 Percent confidence interval - upper         0.079
##   P-value H_0: RMSEA <= 0.050                    0.000
##   P-value H_0: RMSEA >= 0.080                    0.012
##                                                       
##   Robust RMSEA                                   0.076
##   90 Percent confidence interval - lower         0.072
##   90 Percent confidence interval - upper         0.079
##   P-value H_0: Robust RMSEA <= 0.050             0.000
##   P-value H_0: Robust RMSEA >= 0.080             0.027
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.141
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO =~                                                               
##     Im3               1.000                               1.247    0.942
##     Im4               1.049    0.025   42.442    0.000    1.309    0.967
##     Im5               0.802    0.034   23.246    0.000    1.000    0.757
##   FOOD =~                                                               
##     Im10              1.000                               0.821    0.925
##     Im14              1.005    0.035   28.441    0.000    0.825    0.955
##   ATMOS =~                                                              
##     Im20              1.000                               1.255    0.844
##     Im21              0.857    0.042   20.471    0.000    1.075    0.786
##     Im22              1.074    0.046   23.292    0.000    1.347    0.881
##   PRODQUAL =~                                                           
##     Im11              1.000                               0.713    0.619
##     Im12              1.383    0.093   14.904    0.000    0.986    0.872
##     Im13              1.451    0.104   13.906    0.000    1.035    0.859
##   CHOICE =~                                                             
##     Im1               1.000                               1.298    0.977
##     Im2               0.884    0.033   26.613    0.000    1.148    0.898
##   PROF =~                                                               
##     Im16              1.000                               0.929    0.769
##     Im19              1.035    0.058   17.731    0.000    0.962    0.849
##   BRAND =~                                                              
##     Im17              1.000                               1.204    0.968
##     Im18              0.997    0.042   23.544    0.000    1.200    0.857
##   FRENCH =~                                                             
##     Im6               1.000                               0.975    0.810
##     Im7               1.192    0.073   16.341    0.000    1.162    0.961
##   AFCOM =~                                                              
##     COM_A1            1.000                               1.106    0.783
##     COM_A2            1.173    0.056   20.777    0.000    1.298    0.825
##     COM_A3            1.169    0.060   19.433    0.000    1.293    0.814
##     COM_A4            1.306    0.065   20.206    0.000    1.445    0.844
##   SAT =~                                                                
##     SAT_1             1.000                               0.845    0.853
##     SAT_2             0.946    0.050   18.893    0.000    0.799    0.811
##     SAT_3             0.843    0.055   15.435    0.000    0.712    0.643
##   RI =~                                                                 
##     C_REP1            1.000                               0.584    0.791
##     C_REP2            1.031    0.049   21.106    0.000    0.602    0.957
##     C_REP3            0.735    0.039   18.713    0.000    0.429    0.765
##   COI =~                                                                
##     C_CR1             1.000                               1.659    0.847
##     C_CR3             1.029    0.052   19.607    0.000    1.707    0.825
##     C_CR4             0.962    0.050   19.112    0.000    1.596    0.804
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   SAT ~                                                                 
##     DECO             -0.106    0.044   -2.397    0.017   -0.156   -0.156
##     FOOD              0.072    0.073    0.985    0.324    0.070    0.070
##     ATMOS             0.028    0.039    0.698    0.485    0.041    0.041
##     PRODQUAL         -0.028    0.078   -0.355    0.723   -0.023   -0.023
##     CHOICE            0.110    0.042    2.595    0.009    0.169    0.169
##     PROF              0.437    0.094    4.634    0.000    0.480    0.480
##     BRAND             0.014    0.046    0.308    0.758    0.020    0.020
##     FRENCH            0.074    0.058    1.281    0.200    0.086    0.086
##     Im8               0.030    0.054    0.555    0.579    0.036    0.037
##     Im15              0.052    0.046    1.152    0.249    0.062    0.074
##     Im9               0.003    0.033    0.086    0.931    0.003    0.005
##   AFCOM ~                                                               
##     DECO             -0.005    0.056   -0.081    0.935   -0.005   -0.005
##     FOOD              0.027    0.095    0.283    0.778    0.020    0.020
##     ATMOS             0.398    0.054    7.329    0.000    0.451    0.451
##     PRODQUAL         -0.177    0.103   -1.719    0.086   -0.114   -0.114
##     CHOICE            0.087    0.054    1.598    0.110    0.102    0.102
##     PROF              0.048    0.115    0.415    0.678    0.040    0.040
##     BRAND            -0.016    0.060   -0.263    0.792   -0.017   -0.017
##     FRENCH            0.155    0.076    2.040    0.041    0.137    0.137
##     Im8               0.065    0.071    0.912    0.362    0.059    0.061
##     Im15              0.033    0.059    0.556    0.578    0.030    0.035
##     Im9               0.021    0.043    0.479    0.632    0.019    0.026
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   DECO ~~                                                               
##     FOOD              0.424    0.052    8.160    0.000    0.414    0.414
##     ATMOS             0.737    0.084    8.771    0.000    0.471    0.471
##     PRODQUAL          0.419    0.053    7.878    0.000    0.471    0.471
##     CHOICE            0.731    0.081    8.988    0.000    0.452    0.452
##     PROF              0.770    0.074   10.454    0.000    0.664    0.664
##     BRAND             0.785    0.079    9.991    0.000    0.523    0.523
##     FRENCH            0.409    0.066    6.223    0.000    0.336    0.336
##     RI                0.189    0.036    5.257    0.000    0.260    0.260
##     COI               0.084    0.100    0.839    0.402    0.041    0.041
##   FOOD ~~                                                               
##     ATMOS             0.308    0.052    5.894    0.000    0.299    0.299
##     PRODQUAL          0.262    0.035    7.443    0.000    0.448    0.448
##     CHOICE            0.310    0.051    6.055    0.000    0.291    0.291
##     PROF              0.387    0.045    8.532    0.000    0.507    0.507
##     BRAND             0.321    0.048    6.639    0.000    0.325    0.325
##     FRENCH            0.469    0.049    9.620    0.000    0.586    0.586
##     RI                0.126    0.024    5.241    0.000    0.263    0.263
##     COI              -0.033    0.066   -0.491    0.623   -0.024   -0.024
##   ATMOS ~~                                                              
##     PRODQUAL          0.391    0.055    7.134    0.000    0.437    0.437
##     CHOICE            0.753    0.086    8.735    0.000    0.462    0.462
##     PROF              0.558    0.070    7.987    0.000    0.479    0.479
##     BRAND             0.795    0.083    9.628    0.000    0.526    0.526
##     FRENCH            0.408    0.066    6.203    0.000    0.333    0.333
##     RI                0.279    0.041    6.840    0.000    0.380    0.380
##     COI               0.434    0.107    4.044    0.000    0.209    0.209
##   PRODQUAL ~~                                                           
##     CHOICE            0.444    0.055    8.016    0.000    0.480    0.480
##     PROF              0.359    0.045    7.916    0.000    0.542    0.542
##     BRAND             0.482    0.055    8.830    0.000    0.561    0.561
##     FRENCH            0.217    0.039    5.605    0.000    0.313    0.313
##     RI                0.121    0.023    5.337    0.000    0.291    0.291
##     COI               0.075    0.061    1.237    0.216    0.063    0.063
##   CHOICE ~~                                                             
##     PROF              0.732    0.074    9.879    0.000    0.607    0.607
##     BRAND             0.805    0.080   10.024    0.000    0.516    0.516
##     FRENCH            0.271    0.061    4.431    0.000    0.214    0.214
##     RI                0.213    0.038    5.679    0.000    0.281    0.281
##     COI               0.063    0.104    0.602    0.547    0.029    0.029
##   PROF ~~                                                               
##     BRAND             0.674    0.068    9.894    0.000    0.603    0.603
##     FRENCH            0.334    0.053    6.316    0.000    0.368    0.368
##     RI                0.197    0.031    6.354    0.000    0.364    0.364
##     COI              -0.076    0.082   -0.929    0.353   -0.050   -0.050
##   BRAND ~~                                                              
##     FRENCH            0.373    0.063    5.918    0.000    0.318    0.318
##     RI                0.196    0.035    5.619    0.000    0.279    0.279
##     COI               0.125    0.097    1.291    0.197    0.063    0.063
##   FRENCH ~~                                                             
##     RI                0.124    0.030    4.182    0.000    0.218    0.218
##     COI               0.005    0.080    0.064    0.949    0.003    0.003
##   RI ~~                                                                 
##     COI               0.072    0.048    1.498    0.134    0.074    0.074
##  .AFCOM ~~                                                              
##    .SAT               0.205    0.038    5.394    0.000    0.330    0.330
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               4.983    0.058   85.741    0.000    4.983    3.762
##    .Im4               4.995    0.059   84.296    0.000    4.995    3.692
##    .Im5               5.049    0.058   86.540    0.000    5.049    3.823
##    .Im10              6.093    0.039  156.809    0.000    6.093    6.865
##    .Im14              6.139    0.038  161.840    0.000    6.139    7.109
##    .Im20              4.704    0.065   72.277    0.000    4.704    3.166
##    .Im21              5.163    0.060   86.171    0.000    5.163    3.775
##    .Im22              4.306    0.067   63.992    0.000    4.306    2.814
##    .Im11              5.652    0.051  111.591    0.000    5.652    4.907
##    .Im12              5.666    0.050  113.809    0.000    5.666    5.010
##    .Im13              5.450    0.053  103.068    0.000    5.450    4.525
##    .Im1               4.808    0.058   82.567    0.000    4.808    3.618
##    .Im2               4.877    0.056   86.958    0.000    4.877    3.814
##    .Im16              5.150    0.053   96.425    0.000    5.150    4.260
##    .Im19              5.159    0.050  103.705    0.000    5.159    4.551
##    .Im17              5.044    0.054   92.558    0.000    5.044    4.056
##    .Im18              4.603    0.062   74.674    0.000    4.603    3.289
##    .Im6               5.821    0.053  110.235    0.000    5.821    4.835
##    .Im7               5.760    0.053  108.097    0.000    5.760    4.761
##    .COM_A1            3.639    0.533    6.822    0.000    3.639    2.576
##    .COM_A2            3.129    0.626    5.002    0.000    3.129    1.990
##    .COM_A3            2.780    0.623    4.462    0.000    2.780    1.751
##    .COM_A4            2.618    0.695    3.769    0.000    2.618    1.529
##    .SAT_1             4.881    0.409   11.942    0.000    4.881    4.929
##    .SAT_2             5.045    0.385   13.090    0.000    5.045    5.118
##    .SAT_3             5.084    0.346   14.690    0.000    5.084    4.594
##    .C_REP1            4.277    0.032  132.216    0.000    4.277    5.791
##    .C_REP2            4.506    0.028  163.174    0.000    4.506    7.162
##    .C_REP3            4.670    0.025  188.722    0.000    4.670    8.319
##    .C_CR1             2.698    0.086   31.244    0.000    2.698    1.377
##    .C_CR3             3.290    0.091   36.264    0.000    3.290    1.589
##    .C_CR4             2.809    0.087   32.216    0.000    2.809    1.415
##     DECO              0.000                               0.000    0.000
##     FOOD              0.000                               0.000    0.000
##     ATMOS             0.000                               0.000    0.000
##     PRODQUAL          0.000                               0.000    0.000
##     CHOICE            0.000                               0.000    0.000
##     PROF              0.000                               0.000    0.000
##     BRAND             0.000                               0.000    0.000
##     FRENCH            0.000                               0.000    0.000
##    .AFCOM             0.000                               0.000    0.000
##    .SAT               0.000                               0.000    0.000
##     RI                0.000                               0.000    0.000
##     COI               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im3               0.199    0.025    8.074    0.000    0.199    0.113
##    .Im4               0.118    0.025    4.775    0.000    0.118    0.064
##    .Im5               0.744    0.050   14.900    0.000    0.744    0.427
##    .Im10              0.113    0.020    5.810    0.000    0.113    0.144
##    .Im14              0.065    0.019    3.472    0.001    0.065    0.088
##    .Im20              0.634    0.058   10.869    0.000    0.634    0.287
##    .Im21              0.715    0.057   12.589    0.000    0.715    0.382
##    .Im22              0.526    0.060    8.759    0.000    0.526    0.225
##    .Im11              0.818    0.057   14.439    0.000    0.818    0.617
##    .Im12              0.307    0.039    7.804    0.000    0.307    0.240
##    .Im13              0.380    0.045    8.460    0.000    0.380    0.262
##    .Im1               0.080    0.050    1.602    0.109    0.080    0.045
##    .Im2               0.317    0.044    7.258    0.000    0.317    0.194
##    .Im16              0.598    0.051   11.620    0.000    0.598    0.409
##    .Im19              0.360    0.045    8.018    0.000    0.360    0.280
##    .Im17              0.098    0.046    2.118    0.034    0.098    0.064
##    .Im18              0.519    0.056    9.230    0.000    0.519    0.265
##    .Im6               0.498    0.058    8.601    0.000    0.498    0.344
##    .Im7               0.113    0.069    1.633    0.103    0.113    0.077
##    .COM_A1            0.772    0.061   12.623    0.000    0.772    0.387
##    .COM_A2            0.790    0.068   11.538    0.000    0.790    0.319
##    .COM_A3            0.849    0.071   11.957    0.000    0.849    0.337
##    .COM_A4            0.844    0.077   10.980    0.000    0.844    0.288
##    .SAT_1             0.267    0.034    7.954    0.000    0.267    0.272
##    .SAT_2             0.333    0.034    9.833    0.000    0.333    0.343
##    .SAT_3             0.718    0.053   13.633    0.000    0.718    0.586
##    .C_REP1            0.205    0.017   12.097    0.000    0.205    0.375
##    .C_REP2            0.033    0.011    3.004    0.003    0.033    0.085
##    .C_REP3            0.131    0.010   13.529    0.000    0.131    0.415
##    .C_CR1             1.085    0.120    9.059    0.000    1.085    0.283
##    .C_CR3             1.371    0.136   10.115    0.000    1.371    0.320
##    .C_CR4             1.395    0.128   10.868    0.000    1.395    0.354
##     DECO              1.556    0.110   14.096    0.000    1.000    1.000
##     FOOD              0.674    0.052   13.069    0.000    1.000    1.000
##     ATMOS             1.574    0.138   11.430    0.000    1.000    1.000
##     PRODQUAL          0.508    0.070    7.276    0.000    1.000    1.000
##     CHOICE            1.685    0.120   14.027    0.000    1.000    1.000
##     PROF              0.863    0.090    9.575    0.000    1.000    1.000
##     BRAND             1.448    0.106   13.635    0.000    1.000    1.000
##     FRENCH            0.951    0.098    9.751    0.000    1.000    1.000
##    .AFCOM             0.859    0.089    9.681    0.000    0.702    0.702
##    .SAT               0.449    0.048    9.330    0.000    0.629    0.629
##     RI                0.341    0.033   10.256    0.000    1.000    1.000
##     COI               2.752    0.250   11.005    0.000    1.000    1.000
# note: cfa deletes all missing variables in the Xs! (not in Ys) (see help lavOptions)

2.4.3.1 Discussion of global fit measures

Chi square: p-value > 0.05

RMSEA RMSEA <= 0.05 Good fit 0.05 < RMSEA <= 0.08 Acceptable fit 0.08 < RMSEA <= 0.10 Bad fit RMSEA > 0.1 Unacceptable fit

CFI CFI < 0.90 definitely reject model 0.90 < CFI < 0.95 high underrejection rates for misspecified models CFI > 0.95 accept model

2.4.3.2 local fit measures

# semPaths(SEM_fit, what = "path", whatLabels = "std", style = "mx",
#          rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
#          nCharNodes = 7,shapeMan = "rectangle", sizeMan = 8, sizeMan2 = 5, 
#          curvePivot=TRUE, edge.label.cex = 1.2, edge.color = "skyblue4")


semPaths(SEM_fit, what = "path", whatLabels = "std", style = "mx",
         rotation = 2, layout = "tree3", mar = c(1, 2, 1, 2), 
         nCharNodes = 7,shapeMan = "rectangle", 
         sizeMan = 4, sizeMan2 = 3, sizeInt = 2, sizeLat = 6, asize = 1.5,
         curvePivot=TRUE, edge.label.cex = .8, edge.color = "skyblue4"
         )

lambda = inspect(SEM_fit, what="std")$lambda
theta = inspect(SEM_fit, what="std")$theta

# create lambda matrix with ones instead of std.all
ones <- lambda
ones[ones>0] <- 1

# a matrix with dimensions of lambda matrix but with lambdas replaced by thetas
theta_lb <- theta %*% ones
2.4.3.2.1 Indicator reliability criterion
# calculate indicator reliabilities (should be larger than 0.4)
indicrel <- lambda^2/(lambda^2 + theta_lb)
# indicrel

# replace all values satisfying condition with NaN for visibility
indicrel_fail <- indicrel
indicrel_fail[indicrel_fail>.4] <- NaN
indicrel_fail
##        DECO FOOD ATMOS PRODQU CHOICE PROF BRAND FRENCH AFCOM SAT  RI COI Im8
## Im3     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im4     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im5     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im10    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im14    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im20    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im21    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im22    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im11    NaN  NaN   NaN  0.383    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im12    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im13    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im1     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im2     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im16    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im19    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im17    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im18    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im6     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im7     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## COM_A1  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## COM_A2  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## COM_A3  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## COM_A4  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## SAT_1   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## SAT_2   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## SAT_3   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_REP1  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_REP2  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_REP3  NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_CR1   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_CR3   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## C_CR4   NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im8     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im15    NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
## Im9     NaN  NaN   NaN    NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN NaN
##        Im15 Im9
## Im3     NaN NaN
## Im4     NaN NaN
## Im5     NaN NaN
## Im10    NaN NaN
## Im14    NaN NaN
## Im20    NaN NaN
## Im21    NaN NaN
## Im22    NaN NaN
## Im11    NaN NaN
## Im12    NaN NaN
## Im13    NaN NaN
## Im1     NaN NaN
## Im2     NaN NaN
## Im16    NaN NaN
## Im19    NaN NaN
## Im17    NaN NaN
## Im18    NaN NaN
## Im6     NaN NaN
## Im7     NaN NaN
## COM_A1  NaN NaN
## COM_A2  NaN NaN
## COM_A3  NaN NaN
## COM_A4  NaN NaN
## SAT_1   NaN NaN
## SAT_2   NaN NaN
## SAT_3   NaN NaN
## C_REP1  NaN NaN
## C_REP2  NaN NaN
## C_REP3  NaN NaN
## C_CR1   NaN NaN
## C_CR3   NaN NaN
## C_CR4   NaN NaN
## Im8     NaN NaN
## Im15    NaN NaN
## Im9     NaN NaN
2.4.3.2.2 Construct reliability criterion
# calculate construct reliability (should be above .6)
constrrel <- (t(lambda) %*% ones)^2 / ((t(lambda) %*% ones)^2 + t(theta_lb) %*% ones )
# constrrel

# replace all values satisfying condition with NaN for visibility
constrrel_fail <- constrrel
constrrel_fail[constrrel_fail>.6] <- NaN
constrrel_fail
##          DECO FOOD ATMOS PRODQUAL CHOICE PROF BRAND FRENCH AFCOM SAT  RI COI
## DECO      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## FOOD      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## ATMOS     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## PRODQUAL  NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## CHOICE    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## PROF      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## BRAND     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## FRENCH    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## AFCOM     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## SAT       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## RI        NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## COI       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im8       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im15      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im9       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
##          Im8 Im15 Im9
## DECO     NaN  NaN NaN
## FOOD     NaN  NaN NaN
## ATMOS    NaN  NaN NaN
## PRODQUAL NaN  NaN NaN
## CHOICE   NaN  NaN NaN
## PROF     NaN  NaN NaN
## BRAND    NaN  NaN NaN
## FRENCH   NaN  NaN NaN
## AFCOM    NaN  NaN NaN
## SAT      NaN  NaN NaN
## RI       NaN  NaN NaN
## COI      NaN  NaN NaN
## Im8      NaN  NaN NaN
## Im15     NaN  NaN NaN
## Im9      NaN  NaN NaN
2.4.3.2.3 Average Variance Extracted criterion
# calculate Average Variance Extracted (should be above .5)
AVE <- (t(lambda) %*% lambda) / (t(lambda) %*% lambda + t(theta_lb) %*% ones )
# avgvar

# replace all values satisfying condition with NaN for visibility
AVE_fail <- AVE
AVE_fail[AVE_fail>.5] <- NaN
AVE_fail
##          DECO FOOD ATMOS PRODQUAL CHOICE PROF BRAND FRENCH AFCOM SAT  RI COI
## DECO      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## FOOD      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## ATMOS     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## PRODQUAL  NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## CHOICE    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## PROF      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## BRAND     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## FRENCH    NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## AFCOM     NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## SAT       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## RI        NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## COI       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im8       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im15      NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
## Im9       NaN  NaN   NaN      NaN    NaN  NaN   NaN    NaN   NaN NaN NaN NaN
##          Im8 Im15 Im9
## DECO     NaN  NaN NaN
## FOOD     NaN  NaN NaN
## ATMOS    NaN  NaN NaN
## PRODQUAL NaN  NaN NaN
## CHOICE   NaN  NaN NaN
## PROF     NaN  NaN NaN
## BRAND    NaN  NaN NaN
## FRENCH   NaN  NaN NaN
## AFCOM    NaN  NaN NaN
## SAT      NaN  NaN NaN
## RI       NaN  NaN NaN
## COI      NaN  NaN NaN
## Im8      NaN  NaN NaN
## Im15     NaN  NaN NaN
## Im9      NaN  NaN NaN
2.4.3.2.4 Fornell-Larcker Criteria
# correlations between constructs (factors...) should be lower than .7
psi = inspect(SEM_fit, what="std")$psi
psi_fail <- psi
psi_fail[psi_fail<.7] <- NaN
psi_fail
##           DECO  FOOD ATMOS PRODQU CHOICE  PROF BRAND FRENCH AFCOM   SAT    RI
## DECO     1.000                                                               
## FOOD       NaN 1.000                                                         
## ATMOS      NaN   NaN 1.000                                                   
## PRODQUAL   NaN   NaN   NaN  1.000                                            
## CHOICE     NaN   NaN   NaN    NaN  1.000                                     
## PROF       NaN   NaN   NaN    NaN    NaN 1.000                               
## BRAND      NaN   NaN   NaN    NaN    NaN   NaN 1.000                         
## FRENCH     NaN   NaN   NaN    NaN    NaN   NaN   NaN  1.000                  
## AFCOM      NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN 0.702            
## SAT        NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN      
## RI         NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN 1.000
## COI        NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN   NaN
## Im8        NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN   NaN
## Im15       NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN   NaN
## Im9        NaN   NaN   NaN    NaN    NaN   NaN   NaN    NaN   NaN   NaN   NaN
##            COI   Im8  Im15   Im9
## DECO                            
## FOOD                            
## ATMOS                           
## PRODQUAL                        
## CHOICE                          
## PROF                            
## BRAND                           
## FRENCH                          
## AFCOM                           
## SAT                             
## RI                              
## COI      1.000                  
## Im8        NaN 1.000            
## Im15       NaN   NaN 1.000      
## Im9        NaN   NaN   NaN 1.000
# AVE should be higher than squared correlations between constructs

# replace diagonal of psi matrix with AVE values
psi2 <- psi - psi * diag(1,nrow(psi),ncol(psi)) + diag(AVE) * diag(1,nrow(AVE),ncol(AVE))

# create matrix with columns filled with AVE
AVE_full <- AVE
AVE_full[is.na(AVE_full)] <- 0 #replace NAs with 0s
AVE_full <- AVE_full^0 %*% AVE_full # multiply a matrix full of ones with AVE_full to get columns filled with AVE

# substract matrices, replace all values satisfying positive condition (AVE > psi) with NaN
AVEpsi_fail <- AVE_full - psi2
# AVE_full - psi2
AVEpsi_fail[AVEpsi_fail >= 0] <- NaN

AVE_full
##               DECO      FOOD     ATMOS  PRODQUAL   CHOICE      PROF     BRAND
## DECO     0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## FOOD     0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## ATMOS    0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## PRODQUAL 0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## CHOICE   0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## PROF     0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## BRAND    0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## FRENCH   0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## AFCOM    0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## SAT      0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## RI       0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## COI      0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## Im8      0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## Im15     0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
## Im9      0.7986426 0.8841503 0.7020647 0.6269858 0.880352 0.6554196 0.8357675
##             FRENCH    AFCOM       SAT        RI       COI Im8 Im15 Im9
## DECO     0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## FOOD     0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## ATMOS    0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## PRODQUAL 0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## CHOICE   0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## PROF     0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## BRAND    0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## FRENCH   0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## AFCOM    0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## SAT      0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## RI       0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## COI      0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## Im8      0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## Im15     0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
## Im9      0.7895836 0.667259 0.5996322 0.7084962 0.6811628   1    1   1
psi
##            DECO   FOOD  ATMOS PRODQU CHOICE   PROF  BRAND FRENCH  AFCOM    SAT
## DECO      1.000                                                               
## FOOD      0.414  1.000                                                        
## ATMOS     0.471  0.299  1.000                                                 
## PRODQUAL  0.471  0.448  0.437  1.000                                          
## CHOICE    0.452  0.291  0.462  0.480  1.000                                   
## PROF      0.664  0.507  0.479  0.542  0.607  1.000                            
## BRAND     0.523  0.325  0.526  0.561  0.516  0.603  1.000                     
## FRENCH    0.336  0.586  0.333  0.313  0.214  0.368  0.318  1.000              
## AFCOM     0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.702       
## SAT       0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.330  0.629
## RI        0.260  0.263  0.380  0.291  0.281  0.364  0.279  0.218  0.000  0.000
## COI       0.041 -0.024  0.209  0.063  0.029 -0.050  0.063  0.003  0.000  0.000
## Im8       0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000
## Im15      0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000
## Im9       0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000
##              RI    COI    Im8   Im15    Im9
## DECO                                       
## FOOD                                       
## ATMOS                                      
## PRODQUAL                                   
## CHOICE                                     
## PROF                                       
## BRAND                                      
## FRENCH                                     
## AFCOM                                      
## SAT                                        
## RI        1.000                            
## COI       0.074  1.000                     
## Im8       0.000  0.000  1.000              
## Im15      0.000  0.000  0.342  1.000       
## Im9       0.000  0.000  0.442  0.397  1.000
AVEpsi_fail
##            DECO   FOOD  ATMOS PRODQU CHOICE   PROF  BRAND FRENCH  AFCOM    SAT
## DECO          .                                                               
## FOOD        NaN      .                                                        
## ATMOS       NaN    NaN      .                                                 
## PRODQUAL    NaN    NaN    NaN      .                                          
## CHOICE      NaN    NaN    NaN    NaN      .                                   
## PROF        NaN    NaN    NaN    NaN    NaN      .                            
## BRAND       NaN    NaN    NaN    NaN    NaN    NaN      .                     
## FRENCH      NaN    NaN    NaN    NaN    NaN    NaN    NaN      .              
## AFCOM       NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN      .       
## SAT         NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN      .
## RI          NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
## COI         NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
## Im8         NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
## Im15        NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
## Im9         NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN    NaN
##              RI    COI    Im8   Im15    Im9
## DECO                                       
## FOOD                                       
## ATMOS                                      
## PRODQUAL                                   
## CHOICE                                     
## PROF                                       
## BRAND                                      
## FRENCH                                     
## AFCOM                                      
## SAT                                        
## RI            .                            
## COI         NaN      .                     
## Im8         NaN    NaN      .              
## Im15        NaN    NaN    NaN      .       
## Im9         NaN    NaN    NaN    NaN      .

2.4.3.3 Modification indices

arrange(modificationindices(SEM_fit),-mi)
##          lhs op      rhs      mi     epc sepc.lv sepc.all sepc.nox
## 1       Im15  ~   CHOICE 182.859   0.498   0.647    0.543    0.543
## 2        Im8  ~     FOOD 165.501   0.651   0.535    0.509    0.509
## 3       Im15  ~     PROF 145.469   0.660   0.613    0.515    0.515
## 4       Im15  ~      SAT 142.423   1.164   0.983    0.825    0.825
## 5        Im8  ~   FRENCH 131.240   0.492   0.480    0.457    0.457
## 6       Im15  ~    BRAND  87.476   0.374   0.451    0.378    0.378
## 7       Im15  ~ PRODQUAL  85.376   0.649   0.463    0.388    0.388
## 8       Im15  ~     DECO  69.835   0.321   0.400    0.336    0.336
## 9       Im15  ~    ATMOS  69.258   0.329   0.413    0.347    0.347
## 10      Im15  ~    AFCOM  68.913   0.686   0.759    0.637    0.637
## 11     ATMOS ~~    AFCOM  66.719  -1.175  -1.011   -1.011   -1.011
## 12  PRODQUAL ~~    AFCOM  66.255  -3.087  -4.673   -4.673   -4.673
## 13     AFCOM =~   C_REP1  63.992   0.168   0.186    0.252    0.252
## 14      FOOD  ~      Im8  63.636   0.216   0.263    0.276    0.263
## 15     ATMOS  ~    AFCOM  60.988  -1.258  -1.109   -1.109   -1.109
## 16    FRENCH  ~      Im8  60.144   0.268   0.275    0.289    0.275
## 17       Im1 ~~      Im2  51.962  97.454  97.454  611.283  611.283
## 18    FRENCH  ~      Im9  50.101   0.189   0.194    0.263    0.194
## 19    FRENCH  ~    AFCOM  48.545   2.304   2.612    2.612    2.612
## 20    CHOICE  ~     Im15  39.126   0.238   0.183    0.219    0.183
## 21      Im10 ~~     Im14  37.411  32.405  32.405  376.274  376.274
## 22    COM_A1 ~~   COM_A2  36.334   0.316   0.316    0.405    0.405
## 23     AFCOM ~~       RI  33.609   0.139   0.257    0.257    0.257
## 24       Im9  ~    BRAND  31.673   0.245   0.295    0.217    0.217
## 25       Im9  ~   FRENCH  31.190   0.304   0.296    0.218    0.218
## 26     AFCOM ~~      COI  31.130   0.422   0.274    0.274    0.274
## 27       Im8  ~      SAT  29.182   0.453   0.383    0.365    0.365
## 28       Im9  ~ PRODQUAL  27.524   0.401   0.286    0.210    0.210
## 29     ATMOS =~   C_REP1  24.770   0.102   0.128    0.173    0.173
## 30     BRAND ~~      SAT  24.417   6.485   8.039    8.039    8.039
## 31      DECO ~~      SAT  24.321   2.606   3.118    3.118    3.118
## 32      Im16 ~~     Im19  24.136   1.186   1.186    2.556    2.556
## 33      PROF ~~      SAT  23.152  -0.480  -0.771   -0.771   -0.771
## 34       Im8  ~    AFCOM  23.017   0.341   0.377    0.360    0.360
## 35      FOOD ~~      SAT  22.611  -1.744  -3.170   -3.170   -3.170
## 36    CHOICE ~~      SAT  21.514  -8.014  -9.211   -9.211   -9.211
## 37    FRENCH ~~      SAT  21.231  -4.090  -6.257   -6.257   -6.257
## 38    CHOICE ~~    AFCOM  20.712 -10.120  -8.413   -8.413   -8.413
## 39    C_REP2 ~~   C_REP3  19.662   0.093   0.093    1.410    1.410
## 40       SAT =~   C_REP1  19.531   0.126   0.107    0.144    0.144
## 41        RI =~     Im22  19.009  -0.376  -0.219   -0.143   -0.143
## 42     BRAND =~     Im13  18.894   0.196   0.236    0.196    0.196
## 43      Im11 ~~     Im13  18.684  -0.182  -0.182   -0.326   -0.326
## 44       SAT ~~       RI  17.728   0.078   0.200    0.200    0.200
## 45      PROF  ~     Im15  17.174   0.111   0.120    0.143    0.120
## 46        RI =~    SAT_2  16.948   0.234   0.136    0.138    0.138
## 47       Im8  ~     PROF  16.599   0.192   0.178    0.170    0.170
## 48  PRODQUAL  ~      Im9  16.282   0.079   0.111    0.150    0.111
## 49      FOOD ~~    AFCOM  16.106  -1.894  -2.490   -2.490   -2.490
## 50    CHOICE =~     Im20  14.689  -0.154  -0.200   -0.135   -0.135
## 51       SAT =~   COM_A2  14.326  -0.260  -0.220   -0.140   -0.140
## 52     BRAND =~     Im12  14.168  -0.161  -0.194   -0.171   -0.171
## 53    CHOICE  ~      SAT  13.130   2.007   1.306    1.306    1.306
## 54      FOOD =~     Im11  12.888   0.218   0.179    0.155    0.155
## 55    CHOICE =~     Im13  12.504   0.129   0.167    0.139    0.139
## 56    COM_A3 ~~   C_REP1  12.474   0.079   0.079    0.188    0.188
## 57      Im21 ~~   C_REP3  11.833   0.054   0.054    0.178    0.178
## 58       Im9  ~    ATMOS  11.772   0.148   0.186    0.136    0.136
## 59       Im9  ~    AFCOM  11.392   0.304   0.336    0.247    0.247
## 60      Im11 ~~     Im12  11.364   0.135   0.135    0.269    0.269
## 61    CHOICE =~   C_REP1  11.252   0.060   0.078    0.105    0.105
## 62     BRAND  ~      Im9  11.207   0.099   0.082    0.112    0.082
## 63        RI =~   COM_A1  11.088   0.260   0.152    0.108    0.108
## 64     AFCOM =~     Im11  10.927   0.136   0.150    0.131    0.131
## 65     ATMOS  ~      SAT  10.679  -0.764  -0.514   -0.514   -0.514
## 66    FRENCH  ~      SAT  10.672   1.428   1.237    1.237    1.237
## 67       Im8  ~     DECO  10.602   0.108   0.134    0.128    0.128
## 68       SAT =~    C_CR4  10.184   0.249   0.210    0.106    0.106
## 69     ATMOS =~     Im12  10.168  -0.113  -0.142   -0.126   -0.126
## 70      Im13 ~~      Im1  10.148   0.067   0.067    0.384    0.384
## 71  PRODQUAL  ~     Im15  10.056   0.071   0.099    0.118    0.099
## 72      Im21 ~~     Im22   9.808  -0.190  -0.190   -0.310   -0.310
## 73  PRODQUAL =~    C_CR4   9.798   0.291   0.207    0.104    0.104
## 74    CHOICE =~     Im12   9.543  -0.106  -0.138   -0.122   -0.122
## 75     BRAND =~     Im22   9.380   0.147   0.177    0.116    0.116
## 76     BRAND =~     Im20   9.352  -0.143  -0.172   -0.116   -0.116
## 77    FRENCH =~   C_REP1   9.342   0.072   0.070    0.095    0.095
## 78      Im13 ~~     Im17   9.137   0.068   0.068    0.353    0.353
## 79    COM_A1 ~~   COM_A4   8.747  -0.170  -0.170   -0.211   -0.211
## 80     ATMOS =~    C_CR4   8.454   0.156   0.196    0.099    0.099
## 81      Im10 ~~      Im6   8.438  -0.041  -0.041   -0.174   -0.174
## 82       Im4 ~~     Im17   8.347  -0.044  -0.044   -0.413   -0.413
## 83      DECO  ~      SAT   8.162   1.361   0.922    0.922    0.922
## 84      DECO =~   C_REP1   8.000   0.052   0.065    0.088    0.088
## 85      PROF =~   COM_A3   7.907   0.166   0.154    0.097    0.097
## 86    COM_A2 ~~   COM_A3   7.794  -0.167  -0.167   -0.204   -0.204
## 87  PRODQUAL =~    C_CR1   7.757  -0.249  -0.178   -0.091   -0.091
## 88    CHOICE =~   C_REP3   7.675  -0.039  -0.051   -0.091   -0.091
## 89       Im1 ~~    SAT_2   7.648  -0.052  -0.052   -0.319   -0.319
## 90      FOOD =~      Im6   7.597  -0.344  -0.283   -0.235   -0.235
## 91      FOOD =~      Im7   7.597   0.410   0.337    0.279    0.279
## 92      Im10 ~~      Im7   7.537   0.041   0.041    0.359    0.359
## 93       COI =~   C_REP1   7.527   0.038   0.063    0.085    0.085
## 94      Im20 ~~     Im22   7.504   0.206   0.206    0.357    0.357
## 95      DECO =~     Im20   7.348  -0.114  -0.143   -0.096   -0.096
## 96      DECO =~     Im22   7.331   0.117   0.146    0.096    0.096
## 97       SAT =~   COM_A3   7.287   0.190   0.161    0.101    0.101
## 98       SAT ~~      COI   7.250  -0.158  -0.142   -0.142   -0.142
## 99      Im10 ~~     Im16   7.214   0.044   0.044    0.168    0.168
## 100    ATMOS =~   C_REP3   7.085  -0.043  -0.054   -0.096   -0.096
## 101     DECO =~   COM_A3   7.072   0.108   0.135    0.085    0.085
## 102    AFCOM =~    C_CR4   6.939   0.154   0.170    0.086    0.086
## 103    AFCOM =~     Im20   6.866  -0.134  -0.148   -0.099   -0.099
## 104   C_REP1 ~~   C_REP2   6.796  -0.080  -0.080   -0.969   -0.969
## 105     FOOD =~   C_REP1   6.792   0.074   0.060    0.082    0.082
## 106      Im9  ~     DECO   6.779   0.109   0.136    0.100    0.100
## 107     DECO =~    C_CR1   6.728  -0.127  -0.158   -0.081   -0.081
## 108    ATMOS =~     Im11   6.722   0.106   0.133    0.115    0.115
## 109      SAT =~     Im12   6.719  -0.128  -0.108   -0.095   -0.095
## 110     PROF =~   COM_A2   6.718  -0.149  -0.138   -0.088   -0.088
## 111    BRAND =~      Im7   6.517  -0.104  -0.125   -0.103   -0.103
## 112    BRAND =~      Im6   6.517   0.087   0.105    0.087    0.087
## 113      Im3 ~~      Im1   6.441  -0.037  -0.037   -0.297   -0.297
## 114     PROF  ~      SAT   6.393  -0.501  -0.456   -0.456   -0.456
## 115    SAT_2 ~~    SAT_3   6.377  -0.096  -0.096   -0.196   -0.196
## 116     Im22 ~~   C_REP3   6.371  -0.040  -0.040   -0.151   -0.151
## 117   COM_A3 ~~   COM_A4   6.348   0.166   0.166    0.196    0.196
## 118   FRENCH =~     Im22   6.233   0.127   0.124    0.081    0.081
## 119   FRENCH =~   C_REP2   6.230  -0.048  -0.047   -0.075   -0.075
## 120    BRAND =~   COM_A3   6.214   0.106   0.128    0.080    0.080
## 121    BRAND =~      Im4   6.167  -0.068  -0.082   -0.061   -0.061
## 122    BRAND =~    C_CR4   6.120   0.131   0.158    0.079    0.079
## 123     Im22 ~~      Im1   5.965   0.062   0.062    0.302    0.302
## 124     PROF =~    C_CR4   5.960   0.176   0.163    0.082    0.082
## 125 PRODQUAL =~      Im5   5.889   0.169   0.120    0.091    0.091
## 126   CHOICE =~      Im5   5.798   0.085   0.111    0.084    0.084
## 127   COM_A1 ~~    SAT_1   5.734  -0.068  -0.068   -0.150   -0.150
## 128     DECO =~    C_CR4   5.702   0.121   0.151    0.076    0.076
## 129    SAT_1 ~~    SAT_3   5.684   0.096   0.096    0.219    0.219
## 130     Im10 ~~     Im13   5.678  -0.034  -0.034   -0.162   -0.162
## 131     Im19 ~~   C_REP1   5.649  -0.038  -0.038   -0.141   -0.141
## 132     Im22 ~~   C_REP2   5.588  -0.034  -0.034   -0.256   -0.256
## 133       RI =~     Im21   5.555   0.194   0.113    0.083    0.083
## 134     DECO =~      Im7   5.478  -0.093  -0.116   -0.096   -0.096
## 135     DECO =~      Im6   5.478   0.078   0.098    0.081    0.081
## 136     Im20 ~~     Im17   5.475  -0.063  -0.063   -0.253   -0.253
## 137      COI =~    SAT_1   5.406  -0.044  -0.074   -0.075   -0.075
## 138      Im9  ~      SAT   5.377   0.246   0.208    0.153    0.153
## 139   COM_A1 ~~   C_REP2   5.374   0.034   0.034    0.214    0.214
## 140      SAT =~    C_CR1   5.309  -0.172  -0.145   -0.074   -0.074
## 141   FRENCH =~     Im20   5.290  -0.115  -0.112   -0.075   -0.075
## 142    BRAND =~      Im5   5.215   0.093   0.112    0.085    0.085
## 143    ATMOS =~      Im5   5.200   0.089   0.112    0.084    0.084
## 144     Im10 ~~   COM_A2   5.195   0.042   0.042    0.142    0.142
## 145     Im13 ~~     Im16   5.110  -0.068  -0.068   -0.142   -0.142
## 146 PRODQUAL =~      Im1   5.074   0.159   0.113    0.085    0.085
## 147 PRODQUAL =~      Im2   5.074  -0.140  -0.100   -0.078   -0.078
## 148 PRODQUAL  ~    AFCOM   5.056  -0.492  -0.764   -0.764   -0.764
## 149     Im13 ~~      Im2   5.021  -0.047  -0.047   -0.135   -0.135
## 150     FOOD =~   COM_A3   5.013   0.137   0.113    0.071    0.071
## 151   CHOICE =~     Im22   5.005   0.092   0.120    0.078    0.078
## 152     FOOD  ~    AFCOM   4.990   0.539   0.726    0.726    0.726
## 153     PROF =~     Im12   4.951  -0.132  -0.123   -0.108   -0.108
## 154    SAT_3 ~~    C_CR3   4.934  -0.126  -0.126   -0.127   -0.127
## 155     FOOD =~     Im13   4.925  -0.124  -0.102   -0.084   -0.084
## 156      Im3 ~~   COM_A4   4.918   0.057   0.057    0.139    0.139
## 157     Im15  ~     FOOD   4.891   0.130   0.107    0.090    0.090
## 158     Im14 ~~     Im16   4.885  -0.035  -0.035   -0.176   -0.176
## 159      Im1 ~~    SAT_1   4.847   0.041   0.041    0.278    0.278
## 160     Im20 ~~      Im6   4.827  -0.069  -0.069   -0.122   -0.122
## 161     DECO =~     Im18   4.701   0.096   0.120    0.085    0.085
## 162     DECO =~     Im17   4.701  -0.096  -0.120   -0.096   -0.096
## 163     PROF =~   C_REP3   4.690  -0.048  -0.045   -0.079   -0.079
## 164    ATMOS =~   COM_A1   4.664  -0.097  -0.121   -0.086   -0.086
## 165     DECO =~   COM_A2   4.637  -0.085  -0.106   -0.067   -0.067
## 166     Im22 ~~   COM_A1   4.631  -0.085  -0.085   -0.133   -0.133
## 167      Im1 ~~   C_REP3   4.579  -0.022  -0.022   -0.217   -0.217
## 168    AFCOM =~     Im12   4.542  -0.072  -0.080   -0.071   -0.071
## 169     Im20 ~~     Im13   4.512   0.067   0.067    0.136    0.136
## 170      Im4 ~~     Im18   4.472   0.039   0.039    0.158    0.158
## 171     Im13 ~~    SAT_2   4.454   0.049   0.049    0.137    0.137
## 172      Im4 ~~   COM_A3   4.448   0.052   0.052    0.163    0.163
## 173      COI =~   COM_A4   4.402   0.065   0.107    0.063    0.063
## 174     DECO =~   C_REP3   4.388  -0.031  -0.038   -0.068   -0.068
## 175     Im11 ~~      Im6   4.386  -0.066  -0.066   -0.104   -0.104
## 176    BRAND =~    C_CR1   4.338  -0.106  -0.128   -0.065   -0.065
## 177      Im3 ~~      Im4   4.297   0.151   0.151    0.989    0.989
## 178     PROF =~    SAT_2   4.286   0.113   0.105    0.106    0.106
## 179      Im9  ~     FOOD   4.268   0.132   0.109    0.080    0.080
## 180       RI =~     Im16   4.256  -0.174  -0.101   -0.084   -0.084
## 181     Im22 ~~    SAT_3   4.244  -0.077  -0.077   -0.126   -0.126
## 182    AFCOM =~      Im5   4.231   0.082   0.090    0.068    0.068
## 183     Im11 ~~      Im1   4.217  -0.052  -0.052   -0.205   -0.205
## 184    ATMOS =~   C_REP2   4.209  -0.035  -0.044   -0.070   -0.070
## 185    ATMOS =~      Im4   4.208  -0.053  -0.066   -0.049   -0.049
## 186      Im3 ~~      Im5   4.180  -0.067  -0.067   -0.175   -0.175
## 187   FRENCH =~    C_CR4   4.169   0.134   0.131    0.066    0.066
## 188     Im11 ~~   C_REP1   4.127   0.041   0.041    0.100    0.100
## 189     Im20 ~~     Im21   4.109   0.113   0.113    0.168    0.168
## 190     FOOD =~   C_REP2   4.068  -0.047  -0.039   -0.061   -0.061
## 191     FOOD =~    C_CR4   4.061   0.156   0.128    0.065    0.065
## 192     PROF =~   C_REP1   4.060   0.056   0.052    0.071    0.071
## 193      Im7 ~~   C_REP2   3.989  -0.023  -0.023   -0.374   -0.374
## 194     Im14 ~~      Im6   3.978   0.028   0.028    0.155    0.155
## 195   C_REP2 ~~    C_CR4   3.956  -0.042  -0.042   -0.194   -0.194
## 196    BRAND =~   C_REP3   3.918  -0.031  -0.037   -0.066   -0.066
## 197     Im12 ~~      Im7   3.898   0.046   0.046    0.247    0.247
## 198     Im22 ~~    SAT_2   3.894  -0.055  -0.055   -0.132   -0.132
## 199      Im5 ~~      Im6   3.872  -0.060  -0.060   -0.098   -0.098
## 200   COM_A1 ~~    SAT_3   3.837   0.078   0.078    0.104    0.104
## 201     PROF =~    C_CR1   3.832  -0.136  -0.126   -0.064   -0.064
## 202     Im22 ~~     Im11   3.826   0.075   0.075    0.114    0.114
## 203      Im4 ~~      Im6   3.817   0.035   0.035    0.144    0.144
## 204      Im4 ~~     Im11   3.788  -0.043  -0.043   -0.138   -0.138
## 205      Im4 ~~    C_CR1   3.764  -0.059  -0.059   -0.165   -0.165
## 206    BRAND ~~    AFCOM   3.753   3.268   2.931    2.931    2.931
## 207      Im3 ~~      Im2   3.744   0.029   0.029    0.115    0.115
## 208     Im22 ~~   C_REP1   3.729   0.038   0.038    0.115    0.115
## 209     Im14 ~~      Im7   3.715  -0.028  -0.028   -0.328   -0.328
## 210     PROF =~     Im20   3.715  -0.120  -0.111   -0.075   -0.075
## 211       RI =~     Im10   3.676  -0.070  -0.041   -0.046   -0.046
## 212   COM_A1 ~~    C_CR3   3.660   0.113   0.113    0.110    0.110
## 213     Im13 ~~   C_REP2   3.658   0.023   0.023    0.201    0.201
## 214     FOOD =~      Im5   3.653   0.106   0.087    0.066    0.066
## 215     Im20 ~~   COM_A4   3.646   0.084   0.084    0.115    0.115
## 216      Im9  ~     PROF   3.603   0.113   0.105    0.077    0.077
## 217       RI =~      Im5   3.585   0.139   0.081    0.061    0.061
## 218      Im8  ~    ATMOS   3.575   0.064   0.081    0.077    0.077
## 219     DECO  ~    AFCOM   3.564   0.671   0.595    0.595    0.595
## 220    AFCOM =~   C_REP3   3.544  -0.032  -0.035   -0.062   -0.062
## 221 PRODQUAL =~   C_REP3   3.501  -0.051  -0.037   -0.065   -0.065
## 222    BRAND =~    SAT_2   3.488   0.059   0.071    0.072    0.072
## 223     Im16 ~~   C_REP3   3.421  -0.028  -0.028   -0.099   -0.099
## 224   COM_A4 ~~    SAT_1   3.420   0.058   0.058    0.123    0.123
## 225     Im22 ~~    SAT_1   3.404   0.050   0.050    0.134    0.134
## 226     Im10 ~~    C_CR1   3.368   0.044   0.044    0.125    0.125
## 227     Im12 ~~    SAT_3   3.362   0.053   0.053    0.114    0.114
## 228      COI =~   C_REP2   3.326  -0.021  -0.034   -0.054   -0.054
## 229   FRENCH =~   COM_A3   3.325   0.097   0.095    0.060    0.060
## 230      SAT =~     Im13   3.313   0.095   0.080    0.066    0.066
## 231   COM_A2 ~~    SAT_1   3.302  -0.054  -0.054   -0.118   -0.118
## 232      Im8  ~ PRODQUAL   3.293   0.110   0.078    0.075    0.075
## 233     Im13 ~~   COM_A2   3.262  -0.062  -0.062   -0.113   -0.113
## 234      Im3 ~~     Im17   3.252   0.028   0.028    0.201    0.201
## 235     PROF =~    SAT_1   3.232  -0.101  -0.094   -0.095   -0.095
## 236     PROF =~     Im13   3.224   0.112   0.104    0.087    0.087
## 237      SAT =~      Im5   3.190   0.097   0.082    0.062    0.062
## 238      Im2 ~~     Im16   3.139   0.043   0.043    0.099    0.099
## 239      Im5 ~~      Im1   3.128   0.043   0.043    0.177    0.177
## 240     Im11 ~~     Im17   3.123  -0.048  -0.048   -0.168   -0.168
## 241   COM_A4 ~~   C_REP1   3.100   0.040   0.040    0.096    0.096
## 242     PROF =~      Im1   3.078  -0.161  -0.150   -0.113   -0.113
## 243     PROF =~      Im2   3.078   0.142   0.132    0.103    0.103
## 244    SAT_1 ~~    C_CR1   3.074  -0.066  -0.066   -0.123   -0.123
## 245   COM_A1 ~~   COM_A3   3.061  -0.093  -0.093   -0.115   -0.115
## 246   C_REP3 ~~    C_CR3   3.058  -0.041  -0.041   -0.097   -0.097
## 247      COI =~     Im19   3.047   0.048   0.080    0.070    0.070
## 248     Im14 ~~    C_CR1   3.041  -0.040  -0.040   -0.150   -0.150
## 249      Im6 ~~      Im7   3.024  14.806  14.806   62.457   62.457
## 250       RI =~      Im7   2.995  -0.133  -0.078   -0.064   -0.064
## 251     Im14 ~~    SAT_2   2.986   0.021   0.021    0.142    0.142
## 252      Im5 ~~   C_REP1   2.956   0.033   0.033    0.084    0.084
## 253   COM_A2 ~~    C_CR1   2.949   0.100   0.100    0.108    0.108
## 254   COM_A2 ~~    SAT_2   2.932  -0.053  -0.053   -0.103   -0.103
## 255    ATMOS =~   COM_A4   2.915   0.086   0.109    0.063    0.063
## 256      Im2 ~~     Im17   2.894   0.030   0.030    0.172    0.172
## 257      Im2 ~~   C_REP2   2.875  -0.016  -0.016   -0.157   -0.157
## 258      Im5 ~~   COM_A2   2.861   0.068   0.068    0.089    0.089
## 259      Im2 ~~    SAT_2   2.861   0.032   0.032    0.098    0.098
## 260       RI =~     Im14   2.860   0.062   0.036    0.042    0.042
## 261   FRENCH =~     Im19   2.859   0.091   0.089    0.078    0.078
## 262   FRENCH =~     Im16   2.859  -0.088  -0.086   -0.071   -0.071
## 263     Im17 ~~   C_REP1   2.857  -0.023  -0.023   -0.163   -0.163
## 264     Im10 ~~     Im12   2.839   0.022   0.022    0.119    0.119
## 265     Im19 ~~   COM_A1   2.835   0.055   0.055    0.104    0.104
## 266       RI =~     Im12   2.825  -0.110  -0.064   -0.057   -0.057
## 267    BRAND  ~      SAT   2.813   0.820   0.576    0.576    0.576
## 268   CHOICE =~     Im21   2.809   0.065   0.084    0.062    0.062
## 269   C_REP1 ~~   C_REP3   2.788  -0.028  -0.028   -0.173   -0.173
## 270 PRODQUAL =~      Im4   2.787  -0.077  -0.055   -0.041   -0.041
## 271     Im18 ~~   COM_A1   2.759  -0.056  -0.056   -0.089   -0.089
## 272     Im12 ~~      Im6   2.757  -0.040  -0.040   -0.102   -0.102
## 273     FOOD =~    C_CR1   2.748  -0.124  -0.102   -0.052   -0.052
## 274     Im21 ~~    C_CR3   2.739  -0.094  -0.094   -0.095   -0.095
## 275    C_CR1 ~~    C_CR3   2.730   0.715   0.715    0.587    0.587
## 276     Im18 ~~      Im6   2.728   0.044   0.044    0.086    0.086
## 277   CHOICE =~     Im16   2.711   0.084   0.109    0.090    0.090
## 278   CHOICE =~     Im19   2.711  -0.087  -0.113   -0.099   -0.099
## 279      COI =~     Im11   2.705   0.045   0.074    0.065    0.065
## 280   COM_A4 ~~    SAT_3   2.677  -0.072  -0.072   -0.092   -0.092
## 281    ATMOS =~    C_CR1   2.655  -0.085  -0.106   -0.054   -0.054
## 282   FRENCH =~     Im11   2.653   0.078   0.076    0.066    0.066
## 283      SAT =~     Im17   2.634   0.089   0.075    0.060    0.060
## 284     Im13 ~~   C_REP1   2.622  -0.026  -0.026   -0.094   -0.094
## 285      Im7 ~~   COM_A1   2.603  -0.050  -0.050   -0.170   -0.170
## 286      COI =~    SAT_3   2.592  -0.043  -0.071   -0.064   -0.064
## 287     Im22 ~~     Im12   2.573  -0.047  -0.047   -0.116   -0.116
## 288      Im3 ~~     Im11   2.537   0.036   0.036    0.089    0.089
## 289      COI =~     Im22   2.535  -0.046  -0.076   -0.050   -0.050
## 290    AFCOM =~    SAT_2   2.531   0.060   0.066    0.067    0.067
## 291     Im16 ~~    SAT_1   2.523  -0.042  -0.042   -0.106   -0.106
## 292 PRODQUAL ~~      SAT   2.513  -0.466  -0.975   -0.975   -0.975
## 293      Im3 ~~   COM_A3   2.506  -0.040  -0.040   -0.097   -0.097
## 294      Im2 ~~    C_CR1   2.490   0.056   0.056    0.096    0.096
## 295      COI =~      Im1   2.474  -0.032  -0.052   -0.039   -0.039
## 296    ATMOS =~     Im13   2.458   0.059   0.074    0.061    0.061
## 297       RI =~     Im13   2.458   0.109   0.064    0.053    0.053
## 298     Im20 ~~   COM_A1   2.418  -0.062  -0.062   -0.089   -0.089
## 299     Im14 ~~     Im20   2.404  -0.025  -0.025   -0.125   -0.125
## 300     DECO =~     Im12   2.388  -0.055  -0.069   -0.061   -0.061
## 301     Im10 ~~     Im11   2.384   0.027   0.027    0.088    0.088
## 302      Im4 ~~      Im1   2.350   0.022   0.022    0.229    0.229
## 303 PRODQUAL =~    SAT_1   2.347  -0.085  -0.060   -0.061   -0.061
## 304     Im14 ~~     Im12   2.345  -0.019  -0.019   -0.138   -0.138
## 305   COM_A3 ~~   C_REP2   2.330  -0.024  -0.024   -0.145   -0.145
## 306      Im1 ~~     Im17   2.326  -0.028  -0.028   -0.316   -0.316
## 307    BRAND =~   C_REP1   2.319   0.030   0.036    0.048    0.048
## 308   CHOICE =~    C_CR4   2.304   0.074   0.096    0.048    0.048
## 309     Im14 ~~   COM_A2   2.302  -0.027  -0.027   -0.119   -0.119
## 310      Im3 ~~     Im22   2.298   0.033   0.033    0.104    0.104
## 311 PRODQUAL =~    SAT_2   2.278   0.082   0.058    0.059    0.059
## 312   FRENCH =~   COM_A2   2.273  -0.078  -0.076   -0.048   -0.048
## 313      Im2 ~~    SAT_1   2.266  -0.027  -0.027   -0.094   -0.094
## 314     Im10 ~~    SAT_2   2.263  -0.019  -0.019   -0.097   -0.097
## 315    AFCOM =~      Im2   2.255   0.045   0.050    0.039    0.039
## 316   CHOICE =~   COM_A3   2.237   0.059   0.077    0.048    0.048
## 317      COI =~   COM_A2   2.227   0.043   0.072    0.046    0.046
## 318     Im21 ~~    C_CR4   2.224   0.084   0.084    0.084    0.084
## 319      Im2 ~~   C_REP1   2.211   0.020   0.020    0.077    0.077
## 320     Im12 ~~     Im16   2.208   0.042   0.042    0.097    0.097
## 321      Im4 ~~   COM_A4   2.190  -0.037  -0.037   -0.118   -0.118
## 322      COI =~      Im2   2.190   0.026   0.044    0.034    0.034
## 323   FRENCH =~    C_CR1   2.190  -0.094  -0.092   -0.047   -0.047
## 324     Im14 ~~   C_REP3   2.160   0.010   0.010    0.108    0.108
## 325     DECO ~~    AFCOM   2.143   0.995   0.861    0.861    0.861
## 326      SAT =~     Im20   2.138  -0.088  -0.074   -0.050   -0.050
## 327     Im22 ~~     Im13   2.130  -0.046  -0.046   -0.102   -0.102
## 328   COM_A4 ~~    C_CR4   2.110   0.094   0.094    0.086    0.086
## 329     FOOD =~     Im20   2.103  -0.084  -0.069   -0.046   -0.046
## 330      SAT =~     Im18   2.078  -0.079  -0.067   -0.048   -0.048
## 331     Im11 ~~      Im2   2.073   0.037   0.037    0.073    0.073
## 332    AFCOM =~      Im1   2.045  -0.048  -0.054   -0.040   -0.040
## 333    AFCOM =~    SAT_1   2.024  -0.055  -0.061   -0.061   -0.061
## 334     Im14 ~~   COM_A3   1.986   0.026   0.026    0.110    0.110
## 335     Im17 ~~   COM_A1   1.967   0.039   0.039    0.141    0.141
## 336     FOOD =~    SAT_1   1.956  -0.065  -0.053   -0.054   -0.054
## 337    SAT_2 ~~   C_REP1   1.944   0.020   0.020    0.078    0.078
## 338   FRENCH =~   COM_A1   1.940  -0.068  -0.067   -0.047   -0.047
## 339      Im1 ~~   COM_A4   1.929  -0.041  -0.041   -0.156   -0.156
## 340 PRODQUAL =~     Im17   1.902   0.125   0.089    0.072    0.072
## 341 PRODQUAL =~     Im18   1.902  -0.125  -0.089   -0.064   -0.064
## 342       RI =~   COM_A4   1.900   0.120   0.070    0.041    0.041
## 343    SAT_2 ~~    C_CR3   1.885   0.057   0.057    0.085    0.085
## 344      SAT =~     Im11   1.874   0.080   0.068    0.059    0.059
## 345      Im5 ~~     Im16   1.872  -0.048  -0.048   -0.072   -0.072
## 346     PROF =~     Im22   1.847   0.087   0.081    0.053    0.053
## 347     Im18 ~~   C_REP1   1.839   0.023   0.023    0.070    0.070
## 348     Im11 ~~   C_REP2   1.816  -0.020  -0.020   -0.118   -0.118
## 349    ATMOS  ~     Im15   1.808   0.052   0.042    0.050    0.042
## 350    BRAND =~      Im3   1.808   0.036   0.044    0.033    0.033
## 351     Im12 ~~    SAT_1   1.795  -0.028  -0.028   -0.099   -0.099
## 352   C_REP3 ~~    C_CR4   1.794   0.031   0.031    0.073    0.073
## 353     Im20 ~~   C_REP2   1.765   0.019   0.019    0.131    0.131
## 354     Im20 ~~      Im1   1.759  -0.034  -0.034   -0.150   -0.150
## 355    BRAND  ~    AFCOM   1.738   0.489   0.449    0.449    0.449
## 356   COM_A3 ~~   C_REP3   1.735  -0.023  -0.023   -0.070   -0.070
## 357      COI =~    SAT_2   1.725   0.026   0.043    0.043    0.043
## 358     Im12 ~~   COM_A2   1.710   0.042   0.042    0.085    0.085
## 359     Im22 ~~      Im2   1.708  -0.033  -0.033   -0.081   -0.081
## 360    BRAND  ~     Im15   1.699   0.044   0.037    0.044    0.037
## 361      Im2 ~~   C_REP3   1.699   0.014   0.014    0.067    0.067
## 362      Im4 ~~      Im2   1.691  -0.019  -0.019   -0.098   -0.098
## 363      Im5 ~~    SAT_2   1.684   0.035   0.035    0.071    0.071
## 364    AFCOM =~      Im6   1.679   0.048   0.053    0.044    0.044
## 365       RI =~     Im11   1.675   0.102   0.060    0.052    0.052
## 366   CHOICE =~     Im14   1.669   0.021   0.028    0.032    0.032
## 367   CHOICE =~     Im10   1.669  -0.021  -0.028   -0.031   -0.031
## 368   COM_A3 ~~    SAT_2   1.657   0.041   0.041    0.077    0.077
## 369     Im10 ~~     Im20   1.656   0.022   0.022    0.082    0.082
## 370 PRODQUAL =~     Im19   1.649   0.120   0.085    0.075    0.075
## 371 PRODQUAL =~     Im16   1.648  -0.116  -0.083   -0.068   -0.068
## 372     Im11 ~~    SAT_1   1.648   0.035   0.035    0.076    0.076
## 373     Im20 ~~   COM_A2   1.639  -0.053  -0.053   -0.076   -0.076
## 374   C_REP1 ~~    C_CR3   1.636   0.038   0.038    0.071    0.071
## 375     Im13 ~~    SAT_3   1.633  -0.040  -0.040   -0.076   -0.076
## 376    ATMOS =~   COM_A3   1.627   0.062   0.078    0.049    0.049
## 377     Im13 ~~     Im18   1.608  -0.034  -0.034   -0.076   -0.076
## 378      SAT =~      Im2   1.593   0.062   0.052    0.041    0.041
## 379     Im20 ~~    SAT_2   1.578   0.035   0.035    0.077    0.077
## 380 PRODQUAL =~      Im6   1.553  -0.076  -0.054   -0.045   -0.045
## 381 PRODQUAL =~      Im7   1.553   0.090   0.064    0.053    0.053
## 382      COI =~     Im13   1.548  -0.029  -0.048   -0.040   -0.040
## 383     Im14 ~~      Im2   1.521   0.013   0.013    0.093    0.093
## 384     Im19 ~~   C_REP2   1.518   0.015   0.015    0.136    0.136
## 385   FRENCH  ~     Im15   1.500   0.037   0.038    0.046    0.038
## 386     FOOD =~      Im1   1.499  -0.054  -0.044   -0.033   -0.033
## 387     FOOD =~      Im2   1.499   0.048   0.039    0.031    0.031
## 388   FRENCH =~      Im1   1.484  -0.043  -0.042   -0.032   -0.032
## 389   FRENCH =~      Im2   1.484   0.038   0.037    0.029    0.029
## 390     Im21 ~~    SAT_3   1.479   0.046   0.046    0.064    0.064
## 391    SAT_3 ~~   C_REP3   1.471   0.019   0.019    0.062    0.062
## 392      Im7 ~~   COM_A2   1.453  -0.039  -0.039   -0.131   -0.131
## 393     Im20 ~~    SAT_1   1.453  -0.033  -0.033   -0.080   -0.080
## 394     Im10 ~~   C_REP3   1.449  -0.009  -0.009   -0.070   -0.070
## 395     Im22 ~~     Im19   1.448  -0.038  -0.038   -0.088   -0.088
## 396   COM_A1 ~~    SAT_2   1.439   0.035   0.035    0.069    0.069
## 397     Im21 ~~     Im18   1.426  -0.039  -0.039   -0.064   -0.064
## 398      Im3 ~~     Im18   1.403  -0.022  -0.022   -0.070   -0.070
## 399    BRAND =~   COM_A4   1.398  -0.052  -0.063   -0.037   -0.037
## 400   CHOICE =~   COM_A4   1.388  -0.048  -0.062   -0.036   -0.036
## 401      Im2 ~~     Im18   1.384  -0.025  -0.025   -0.063   -0.063
## 402    BRAND =~    SAT_1   1.384  -0.038  -0.046   -0.046   -0.046
## 403      Im5 ~~      Im7   1.347   0.033   0.033    0.115    0.115
## 404      Im2 ~~   COM_A3   1.341   0.033   0.033    0.064    0.064
## 405   COM_A1 ~~   C_REP3   1.341   0.019   0.019    0.060    0.060
## 406      Im7 ~~    SAT_3   1.336   0.034   0.034    0.120    0.120
## 407     Im22 ~~    C_CR1   1.324  -0.061  -0.061   -0.081   -0.081
## 408     Im20 ~~    SAT_3   1.321   0.044   0.044    0.065    0.065
## 409    BRAND =~   COM_A2   1.318  -0.047  -0.057   -0.036   -0.036
## 410      COI =~      Im4   1.314  -0.018  -0.030   -0.022   -0.022
## 411     DECO =~     Im13   1.308   0.043   0.054    0.044    0.044
## 412     Im12 ~~    C_CR3   1.299   0.050   0.050    0.077    0.077
## 413     Im12 ~~   COM_A4   1.299  -0.039  -0.039   -0.076   -0.076
## 414      Im4 ~~    C_CR3   1.288   0.037   0.037    0.092    0.092
## 415   COM_A3 ~~    SAT_3   1.286  -0.049  -0.049   -0.062   -0.062
## 416    SAT_1 ~~    C_CR3   1.274  -0.046  -0.046   -0.076   -0.076
## 417     Im12 ~~   C_REP1   1.266  -0.017  -0.017   -0.068   -0.068
## 418    ATMOS ~~      SAT   1.253  -0.125  -0.148   -0.148   -0.148
## 419     Im12 ~~     Im17   1.253  -0.024  -0.024   -0.136   -0.136
## 420      Im6 ~~   COM_A1   1.240   0.036   0.036    0.059    0.059
## 421       RI =~      Im1   1.236  -0.068  -0.040   -0.030   -0.030
## 422    ATMOS =~    C_CR3   1.228  -0.061  -0.077   -0.037   -0.037
## 423     Im20 ~~     Im19   1.227   0.035   0.035    0.074    0.074
## 424      SAT =~     Im21   1.217   0.065   0.055    0.040    0.040
## 425 PRODQUAL =~   COM_A2   1.207  -0.078  -0.055   -0.035   -0.035
## 426      SAT =~    C_CR3   1.207  -0.087  -0.074   -0.036   -0.036
## 427    ATMOS =~     Im17   1.181  -0.050  -0.063   -0.051   -0.051
## 428    ATMOS =~     Im18   1.181   0.050   0.063    0.045    0.045
## 429     PROF =~      Im7   1.169  -0.067  -0.062   -0.052   -0.052
## 430     PROF =~      Im6   1.169   0.056   0.052    0.044    0.044
## 431    BRAND  ~      Im8   1.166  -0.041  -0.034   -0.036   -0.034
## 432     Im19 ~~    SAT_3   1.165  -0.034  -0.034   -0.066   -0.066
## 433     Im14 ~~     Im13   1.160   0.015   0.015    0.093    0.093
## 434     Im13 ~~    C_CR3   1.150  -0.050  -0.050   -0.070   -0.070
## 435      Im6 ~~   C_REP2   1.148   0.013   0.013    0.097    0.097
## 436      Im5 ~~     Im14   1.144   0.017   0.017    0.077    0.077
## 437     Im16 ~~    C_CR4   1.143  -0.057  -0.057   -0.062   -0.062
## 438      Im3 ~~    C_CR1   1.139   0.033   0.033    0.072    0.072
## 439      Im1 ~~   COM_A3   1.137  -0.030  -0.030   -0.116   -0.116
## 440    ATMOS =~   COM_A2   1.135  -0.051  -0.063   -0.040   -0.040
## 441      Im7 ~~   COM_A4   1.132   0.037   0.037    0.119    0.119
## 442     Im16 ~~    C_CR1   1.132   0.054   0.054    0.067    0.067
## 443 PRODQUAL =~   C_REP2   1.125   0.030   0.022    0.034    0.034
## 444     Im21 ~~     Im11   1.123  -0.041  -0.041   -0.054   -0.054
## 445     Im14 ~~     Im17   1.121   0.012   0.012    0.153    0.153
## 446      Im5 ~~      Im2   1.115  -0.026  -0.026   -0.054   -0.054
## 447      SAT =~     Im14   1.111   0.031   0.026    0.030    0.030
## 448     Im12 ~~    SAT_2   1.111  -0.023  -0.023   -0.071   -0.071
## 449    SAT_1 ~~    C_CR4   1.107   0.042   0.042    0.069    0.069
## 450     Im11 ~~      Im7   1.106   0.032   0.032    0.104    0.104
## 451     Im12 ~~     Im13   1.104   0.077   0.077    0.226    0.226
## 452      SAT =~     Im10   1.103  -0.030  -0.026   -0.029   -0.029
## 453     Im17 ~~   C_REP2   1.100   0.011   0.011    0.188    0.188
## 454      Im3 ~~     Im12   1.084  -0.018  -0.018   -0.072   -0.072
## 455   FRENCH ~~    AFCOM   1.083   1.188   1.314    1.314    1.314
## 456     Im17 ~~   COM_A4   1.064  -0.032  -0.032   -0.110   -0.110
## 457       RI =~   COM_A3   1.052   0.087   0.051    0.032    0.032
## 458     PROF =~      Im4   1.046  -0.050  -0.047   -0.034   -0.034
## 459     Im18 ~~   C_REP3   1.035  -0.014  -0.014   -0.052   -0.052
## 460     Im19 ~~      Im7   1.026   0.027   0.027    0.133    0.133
## 461      Im3 ~~   COM_A2   1.026  -0.025  -0.025   -0.062   -0.062
## 462 PRODQUAL =~   COM_A3   1.023   0.073   0.052    0.033    0.033
## 463     Im11 ~~   COM_A3   1.020   0.044   0.044    0.053    0.053
## 464     Im18 ~~   COM_A2   1.018   0.036   0.036    0.056    0.056
## 465     Im22 ~~     Im18   1.017   0.033   0.033    0.062    0.062
## 466    ATMOS  ~      Im9   1.016   0.034   0.027    0.037    0.027
## 467     Im10 ~~     Im17   1.010  -0.012  -0.012   -0.114   -0.114
## 468   COM_A2 ~~   COM_A4   1.008  -0.066  -0.066   -0.081   -0.081
## 469     Im22 ~~      Im7   0.994   0.030   0.030    0.123    0.123
## 470     Im12 ~~    C_CR1   0.992  -0.041  -0.041   -0.071   -0.071
## 471     PROF  ~      Im8   0.991   0.030   0.033    0.034    0.033
## 472     Im19 ~~    SAT_2   0.986   0.024   0.024    0.068    0.068
## 473      Im1 ~~     Im18   0.978   0.021   0.021    0.104    0.104
## 474      Im3 ~~    SAT_1   0.972   0.016   0.016    0.068    0.068
## 475    AFCOM =~     Im16   0.967  -0.044  -0.049   -0.040   -0.040
## 476     Im12 ~~   COM_A3   0.964  -0.032  -0.032   -0.063   -0.063
## 477     Im11 ~~    SAT_2   0.963  -0.028  -0.028   -0.054   -0.054
## 478      Im6 ~~    SAT_1   0.962   0.022   0.022    0.060    0.060
## 479     Im13 ~~      Im7   0.946  -0.024  -0.024   -0.117   -0.117
## 480     Im18 ~~      Im7   0.944  -0.024  -0.024   -0.101   -0.101
## 481   COM_A2 ~~    SAT_3   0.942   0.040   0.040    0.053    0.053
## 482    SAT_1 ~~   C_REP3   0.932   0.011   0.011    0.058    0.058
## 483     FOOD =~    SAT_2   0.929   0.044   0.036    0.037    0.037
## 484      SAT =~      Im4   0.927  -0.033  -0.028   -0.021   -0.021
## 485   FRENCH =~     Im13   0.926  -0.041  -0.040   -0.033   -0.033
## 486     Im11 ~~    C_CR4   0.924   0.055   0.055    0.051    0.051
## 487     Im19 ~~     Im17   0.923   0.023   0.023    0.121    0.121
## 488   FRENCH =~      Im5   0.920   0.043   0.042    0.032    0.032
## 489      Im1 ~~   C_REP2   0.910   0.009   0.009    0.178    0.178
## 490     Im20 ~~     Im18   0.908   0.031   0.031    0.054    0.054
## 491   FRENCH =~   COM_A4   0.907   0.052   0.051    0.030    0.030
## 492      Im4 ~~     Im22   0.874  -0.020  -0.020   -0.081   -0.081
## 493    ATMOS =~      Im3   0.867   0.024   0.029    0.022    0.022
## 494     PROF =~      Im5   0.861   0.063   0.058    0.044    0.044
## 495      Im7 ~~   C_REP3   0.861   0.011   0.011    0.093    0.093
## 496      Im5 ~~     Im19   0.860  -0.028  -0.028   -0.054   -0.054
## 497      COI =~      Im5   0.858   0.024   0.040    0.030    0.030
## 498     FOOD =~     Im22   0.845   0.054   0.044    0.029    0.029
## 499     Im13 ~~      Im6   0.841   0.024   0.024    0.054    0.054
## 500     Im17 ~~      Im7   0.837  -0.020  -0.020   -0.191   -0.191
## 501     Im12 ~~    C_CR4   0.836   0.039   0.039    0.060    0.060
## 502      Im4 ~~      Im7   0.830  -0.016  -0.016   -0.135   -0.135
## 503    BRAND =~    SAT_3   0.826  -0.036  -0.043   -0.039   -0.039
## 504     FOOD  ~      Im9   0.825   0.019   0.023    0.031    0.023
## 505   COM_A4 ~~    SAT_2   0.818   0.029   0.029    0.056    0.056
## 506     Im22 ~~     Im16   0.817   0.033   0.033    0.058    0.058
## 507     Im20 ~~   C_REP3   0.816  -0.014  -0.014   -0.050   -0.050
## 508   CHOICE =~    C_CR1   0.811  -0.042  -0.055   -0.028   -0.028
## 509     Im10 ~~   COM_A3   0.806  -0.017  -0.017   -0.056   -0.056
## 510   COM_A1 ~~   C_REP1   0.804  -0.018  -0.018   -0.046   -0.046
## 511      SAT =~      Im1   0.803  -0.050  -0.042   -0.032   -0.032
## 512    BRAND =~     Im19   0.798   0.051   0.062    0.054    0.054
## 513    BRAND =~     Im16   0.798  -0.049  -0.059   -0.049   -0.049
## 514     Im16 ~~   COM_A4   0.793  -0.037  -0.037   -0.053   -0.053
## 515     Im11 ~~     Im18   0.790   0.029   0.029    0.045    0.045
## 516     Im18 ~~    SAT_1   0.788  -0.021  -0.021   -0.055   -0.055
## 517     Im14 ~~     Im21   0.786   0.015   0.015    0.067    0.067
## 518      Im9  ~   CHOICE   0.783   0.035   0.046    0.034    0.034
## 519     PROF =~     Im17   0.779   0.076   0.071    0.057    0.057
## 520     PROF =~     Im18   0.779  -0.076  -0.071   -0.051   -0.051
## 521      Im7 ~~    SAT_2   0.768  -0.019  -0.019   -0.101   -0.101
## 522      Im4 ~~      Im5   0.764   0.030   0.030    0.103    0.103
## 523     Im10 ~~   COM_A4   0.759  -0.017  -0.017   -0.055   -0.055
## 524     Im11 ~~   COM_A2   0.754   0.037   0.037    0.046    0.046
## 525     FOOD =~   COM_A2   0.752  -0.052  -0.042   -0.027   -0.027
## 526     DECO =~    SAT_2   0.751   0.025   0.031    0.032    0.032
## 527     Im10 ~~   COM_A1   0.740  -0.015  -0.015   -0.052   -0.052
## 528      Im7 ~~   COM_A3   0.732   0.029   0.029    0.093    0.093
## 529    C_CR3 ~~    C_CR4   0.727  -0.331  -0.331   -0.240   -0.240
## 530   CHOICE  ~    AFCOM   0.725   0.358   0.305    0.305    0.305
## 531   FRENCH =~    SAT_3   0.721   0.040   0.039    0.035    0.035
## 532      Im7 ~~    C_CR4   0.719   0.037   0.037    0.094    0.094
## 533   CHOICE =~   COM_A2   0.719  -0.033  -0.042   -0.027   -0.027
## 534     Im16 ~~   COM_A3   0.716   0.035   0.035    0.049    0.049
## 535     Im13 ~~   COM_A3   0.708  -0.030  -0.030   -0.052   -0.052
## 536     Im13 ~~   COM_A4   0.705   0.030   0.030    0.054    0.054
## 537    SAT_2 ~~   C_REP2   0.703   0.009   0.009    0.083    0.083
## 538    BRAND =~     Im11   0.700  -0.038  -0.046   -0.040   -0.040
## 539      Im5 ~~     Im11   0.698   0.031   0.031    0.040    0.040
## 540     Im20 ~~   COM_A3   0.689  -0.036  -0.036   -0.049   -0.049
## 541     Im16 ~~    SAT_3   0.689   0.030   0.030    0.046    0.046
## 542     Im12 ~~      Im2   0.681  -0.016  -0.016   -0.052   -0.052
## 543    SAT_2 ~~   C_REP3   0.681   0.010   0.010    0.046    0.046
## 544     Im22 ~~      Im6   0.676   0.025   0.025    0.050    0.050
## 545    ATMOS =~      Im2   0.673  -0.028  -0.035   -0.027   -0.027
## 546    ATMOS =~      Im1   0.673   0.031   0.039    0.029    0.029
## 547     Im10 ~~    C_CR4   0.670  -0.021  -0.021   -0.052   -0.052
## 548      Im1 ~~    C_CR1   0.670  -0.029  -0.029   -0.098   -0.098
## 549     Im22 ~~     Im17   0.669   0.022   0.022    0.097    0.097
## 550    ATMOS =~    SAT_2   0.668   0.025   0.031    0.032    0.032
## 551     Im14 ~~    C_CR4   0.666   0.020   0.020    0.065    0.065
## 552     Im16 ~~   COM_A1   0.665  -0.031  -0.031   -0.045   -0.045
## 553     Im17 ~~   COM_A3   0.664   0.024   0.024    0.085    0.085
## 554    AFCOM =~     Im19   0.663   0.037   0.041    0.036    0.036
## 555     Im10 ~~    C_CR3   0.654  -0.021  -0.021   -0.052   -0.052
## 556      Im5 ~~     Im22   0.651   0.029   0.029    0.047    0.047
## 557   COM_A3 ~~    C_CR3   0.646   0.052   0.052    0.048    0.048
## 558    BRAND =~     Im14   0.641   0.015   0.018    0.021    0.021
## 559    BRAND =~     Im10   0.641  -0.015  -0.018   -0.020   -0.020
## 560    AFCOM =~     Im14   0.628   0.015   0.017    0.020    0.020
## 561    SAT_3 ~~    C_CR1   0.614   0.042   0.042    0.047    0.047
## 562      Im5 ~~    C_CR4   0.609   0.042   0.042    0.042    0.042
## 563     Im19 ~~    SAT_1   0.608  -0.018  -0.018   -0.059   -0.059
## 564     Im20 ~~      Im2   0.603  -0.020  -0.020   -0.044   -0.044
## 565      Im4 ~~     Im16   0.603   0.016   0.016    0.062    0.062
## 566      COI =~     Im12   0.592   0.017   0.028    0.025    0.025
## 567     Im17 ~~   C_REP3   0.584   0.008   0.008    0.074    0.074
## 568     Im19 ~~     Im18   0.583  -0.020  -0.020   -0.047   -0.047
## 569   COM_A4 ~~   C_REP3   0.582  -0.014  -0.014   -0.042   -0.042
## 570     Im14 ~~    C_CR3   0.579   0.019   0.019    0.062    0.062
## 571       RI =~      Im6   0.578   0.049   0.029    0.024    0.024
## 572      Im1 ~~   COM_A1   0.573   0.020   0.020    0.080    0.080
## 573     Im14 ~~   C_REP1   0.570   0.006   0.006    0.055    0.055
## 574     Im14 ~~     Im18   0.569  -0.010  -0.010   -0.057   -0.057
## 575      COI =~     Im16   0.562  -0.021  -0.034   -0.028   -0.028
## 576      Im5 ~~   C_REP2   0.559  -0.010  -0.010   -0.065   -0.065
## 577      Im6 ~~    C_CR4   0.557  -0.034  -0.034   -0.041   -0.041
## 578     FOOD =~      Im4   0.554  -0.027  -0.022   -0.016   -0.016
## 579     Im16 ~~      Im7   0.553  -0.022  -0.022   -0.083   -0.083
## 580       RI =~      Im4   0.552  -0.034  -0.020   -0.015   -0.015
## 581      COI =~   COM_A3   0.549   0.022   0.037    0.023    0.023
## 582     Im21 ~~   COM_A4   0.549  -0.033  -0.033   -0.042   -0.042
## 583     Im13 ~~     Im19   0.549   0.019   0.019    0.053    0.053
## 584     Im14 ~~    SAT_1   0.542  -0.009  -0.009   -0.066   -0.066
## 585     PROF =~   COM_A4   0.539  -0.045  -0.042   -0.024   -0.024
## 586      Im4 ~~     Im12   0.538   0.012   0.012    0.064    0.064
## 587     Im17 ~~      Im6   0.519   0.016   0.016    0.071    0.071
## 588     Im17 ~~    SAT_3   0.519  -0.019  -0.019   -0.072   -0.072
## 589     DECO =~   C_REP2   0.514  -0.011  -0.014   -0.022   -0.022
## 590     FOOD =~   COM_A1   0.511  -0.040  -0.033   -0.023   -0.023
## 591      Im4 ~~   C_REP3   0.508  -0.006  -0.006   -0.052   -0.052
## 592      Im1 ~~   COM_A2   0.501   0.020   0.020    0.078    0.078
## 593    C_CR1 ~~    C_CR4   0.501  -0.275  -0.275   -0.223   -0.223
## 594     PROF =~   COM_A1   0.495   0.038   0.035    0.025    0.025
## 595     Im20 ~~      Im7   0.492   0.021   0.021    0.079    0.079
## 596      Im6 ~~   COM_A3   0.490  -0.025  -0.025   -0.038   -0.038
## 597   CHOICE =~   COM_A1   0.490   0.025   0.033    0.023    0.023
## 598       RI =~     Im17   0.487   0.048   0.028    0.023    0.023
## 599   C_REP2 ~~    C_CR3   0.482   0.015   0.015    0.069    0.069
## 600     Im17 ~~    SAT_2   0.482   0.014   0.014    0.076    0.076
## 601      SAT =~      Im6   0.477   0.035   0.030    0.025    0.025
## 602     Im21 ~~   C_REP1   0.477  -0.014  -0.014   -0.036   -0.036
## 603 PRODQUAL =~   COM_A1   0.476   0.046   0.033    0.023    0.023
## 604     Im11 ~~   COM_A1   0.474  -0.028  -0.028   -0.035   -0.035
## 605     DECO  ~     Im15   0.472   0.024   0.019    0.023    0.019
## 606      Im6 ~~    SAT_3   0.469  -0.021  -0.021   -0.036   -0.036
## 607      Im2 ~~   COM_A4   0.468   0.020   0.020    0.039    0.039
## 608       RI =~     Im18   0.466  -0.047  -0.027   -0.020   -0.020
## 609   CHOICE =~      Im3   0.466  -0.016  -0.020   -0.015   -0.015
## 610      Im6 ~~   COM_A2   0.466   0.023   0.023    0.037    0.037
## 611     Im12 ~~   C_REP2   0.462   0.008   0.008    0.074    0.074
## 612     Im18 ~~   C_REP2   0.462  -0.008  -0.008   -0.063   -0.063
## 613     Im10 ~~     Im18   0.462   0.010   0.010    0.041    0.041
## 614     FOOD =~    SAT_3   0.459   0.039   0.032    0.029    0.029
## 615     Im20 ~~   C_REP1   0.456  -0.013  -0.013   -0.037   -0.037
## 616     Im21 ~~     Im17   0.454   0.018   0.018    0.068    0.068
## 617    BRAND =~      Im1   0.454  -0.030  -0.036   -0.027   -0.027
## 618    BRAND =~      Im2   0.454   0.026   0.032    0.025    0.025
## 619   COM_A3 ~~    SAT_1   0.450   0.021   0.021    0.043    0.043
## 620      Im2 ~~      Im7   0.443   0.013   0.013    0.069    0.069
## 621   CHOICE =~     Im11   0.442  -0.026  -0.034   -0.029   -0.029
## 622      Im7 ~~    C_CR1   0.434  -0.028  -0.028   -0.079   -0.079
## 623    AFCOM =~   C_REP2   0.431  -0.010  -0.011   -0.018   -0.018
## 624      Im3 ~~     Im20   0.430  -0.015  -0.015   -0.041   -0.041
## 625      Im5 ~~     Im21   0.427  -0.024  -0.024   -0.033   -0.033
## 626   COM_A1 ~~    C_CR1   0.421  -0.036  -0.036   -0.039   -0.039
## 627   COM_A4 ~~    C_CR3   0.420  -0.043  -0.043   -0.040   -0.040
## 628     PROF =~     Im11   0.413   0.041   0.038    0.033    0.033
## 629     Im10 ~~      Im2   0.411  -0.007  -0.007   -0.038   -0.038
## 630     DECO =~     Im11   0.410   0.026   0.032    0.028    0.028
## 631     Im10 ~~    SAT_3   0.410   0.011   0.011    0.038    0.038
## 632       RI =~   COM_A2   0.409  -0.052  -0.031   -0.019   -0.019
## 633     Im12 ~~   COM_A1   0.408   0.019   0.019    0.040    0.040
## 634   CHOICE =~   C_REP2   0.408  -0.009  -0.012   -0.019   -0.019
## 635     Im21 ~~    SAT_2   0.407  -0.018  -0.018   -0.037   -0.037
## 636      Im3 ~~      Im7   0.407  -0.011  -0.011   -0.074   -0.074
## 637     Im13 ~~   COM_A1   0.405   0.021   0.021    0.038    0.038
## 638   CHOICE =~    SAT_2   0.402  -0.020  -0.026   -0.026   -0.026
## 639      Im3 ~~     Im19   0.398   0.012   0.012    0.044    0.044
## 640     FOOD =~     Im19   0.396   0.045   0.037    0.033    0.033
## 641     FOOD =~     Im16   0.396  -0.044  -0.036   -0.030   -0.030
## 642   CHOICE  ~      Im8   0.396  -0.027  -0.021   -0.022   -0.021
## 643     FOOD =~   COM_A4   0.395  -0.040  -0.033   -0.019   -0.019
## 644      Im1 ~~    SAT_3   0.390   0.016   0.016    0.065    0.065
## 645    AFCOM =~     Im10   0.378  -0.012  -0.013   -0.015   -0.015
## 646      SAT =~     Im16   0.375  -0.047  -0.039   -0.033   -0.033
## 647     Im16 ~~   C_REP1   0.375   0.011   0.011    0.033    0.033
## 648      SAT =~   COM_A4   0.375   0.045   0.038    0.022    0.022
## 649    AFCOM =~      Im4   0.364  -0.015  -0.017   -0.012   -0.012
## 650     Im22 ~~   COM_A3   0.361   0.026   0.026    0.038    0.038
## 651    SAT_1 ~~   C_REP2   0.361  -0.006  -0.006   -0.064   -0.064
## 652     Im21 ~~   COM_A1   0.353   0.024   0.024    0.032    0.032
## 653    SAT_2 ~~    C_CR1   0.348  -0.023  -0.023   -0.038   -0.038
## 654      Im6 ~~   C_REP3   0.343  -0.008  -0.008   -0.030   -0.030
## 655      SAT =~   COM_A1   0.342   0.038   0.032    0.023    0.023
## 656     DECO =~    SAT_3   0.330  -0.021  -0.026   -0.024   -0.024
## 657      Im5 ~~    C_CR3   0.325  -0.031  -0.031   -0.031   -0.031
## 658     FOOD  ~     Im15   0.324  -0.014  -0.017   -0.020   -0.017
## 659     Im20 ~~    C_CR3   0.323  -0.032  -0.032   -0.035   -0.035
## 660     PROF =~     Im21   0.323   0.034   0.031    0.023    0.023
## 661      COI =~      Im3   0.321   0.009   0.015    0.011    0.011
## 662     Im14 ~~   C_REP2   0.319  -0.004  -0.004   -0.075   -0.075
## 663    ATMOS =~     Im14   0.318   0.010   0.013    0.015    0.015
## 664    ATMOS =~     Im10   0.318  -0.010  -0.013   -0.014   -0.014
## 665     Im17 ~~    C_CR1   0.310  -0.021  -0.021   -0.064   -0.064
## 666     Im11 ~~   COM_A4   0.305   0.025   0.025    0.030    0.030
## 667      Im5 ~~   COM_A4   0.305  -0.024  -0.024   -0.030   -0.030
## 668      Im4 ~~   C_REP2   0.305   0.005   0.005    0.072    0.072
## 669      Im5 ~~     Im17   0.301   0.014   0.014    0.052    0.052
## 670     FOOD =~     Im21   0.300   0.031   0.025    0.018    0.018
## 671   CHOICE =~      Im4   0.295  -0.013  -0.016   -0.012   -0.012
## 672 PRODQUAL =~     Im22   0.295  -0.042  -0.030   -0.020   -0.020
## 673     Im21 ~~      Im1   0.293   0.014   0.014    0.057    0.057
## 674     Im19 ~~   COM_A2   0.293  -0.019  -0.019   -0.035   -0.035
## 675 PRODQUAL =~     Im20   0.286   0.041   0.029    0.019    0.019
## 676      Im4 ~~   COM_A2   0.283  -0.013  -0.013   -0.041   -0.041
## 677    ATMOS =~    SAT_3   0.282  -0.020  -0.025   -0.023   -0.023
## 678     Im21 ~~   COM_A2   0.282   0.022   0.022    0.029    0.029
## 679     Im12 ~~   C_REP3   0.280  -0.006  -0.006   -0.032   -0.032
## 680      Im7 ~~   C_REP1   0.278   0.008   0.008    0.053    0.053
## 681   CHOICE =~    C_CR3   0.274  -0.026  -0.034   -0.016   -0.016
## 682   COM_A4 ~~    C_CR1   0.274   0.032   0.032    0.034    0.034
## 683      Im2 ~~   COM_A1   0.273  -0.014  -0.014   -0.028   -0.028
## 684   CHOICE =~     Im18   0.271  -0.022  -0.028   -0.020   -0.020
## 685   CHOICE =~     Im17   0.271   0.022   0.028    0.023    0.023
## 686   COM_A1 ~~    C_CR4   0.271  -0.030  -0.030   -0.029   -0.029
## 687   FRENCH =~      Im3   0.270  -0.015  -0.014   -0.011   -0.011
## 688     PROF =~      Im3   0.269   0.025   0.023    0.017    0.017
## 689     Im14 ~~     Im22   0.265   0.008   0.008    0.045    0.045
## 690     Im17 ~~   COM_A2   0.264  -0.015  -0.015   -0.054   -0.054
## 691    AFCOM =~      Im7   0.263  -0.023  -0.025   -0.021   -0.021
## 692       RI =~    SAT_1   0.262  -0.028  -0.017   -0.017   -0.017
## 693     Im21 ~~     Im16   0.262  -0.019  -0.019   -0.028   -0.028
## 694     Im16 ~~    C_CR3   0.255  -0.027  -0.027   -0.030   -0.030
## 695    AFCOM =~     Im17   0.254   0.019   0.021    0.017    0.017
## 696      Im6 ~~    C_CR1   0.252   0.022   0.022    0.030    0.030
## 697      Im8  ~   CHOICE   0.251  -0.016  -0.021   -0.020   -0.020
## 698     Im10 ~~     Im21   0.249  -0.009  -0.009   -0.030   -0.030
## 699     Im14 ~~     Im11   0.249   0.008   0.008    0.036    0.036
## 700    AFCOM =~     Im21   0.246   0.024   0.027    0.020    0.020
## 701     Im21 ~~      Im7   0.245  -0.015  -0.015   -0.052   -0.052
## 702      Im1 ~~      Im6   0.239  -0.010  -0.010   -0.051   -0.051
## 703      Im5 ~~    SAT_1   0.239  -0.013  -0.013   -0.029   -0.029
## 704     Im19 ~~    C_CR3   0.237   0.023   0.023    0.033    0.033
## 705     PROF  ~    AFCOM   0.233  -0.066  -0.079   -0.079   -0.079
## 706 PRODQUAL =~   COM_A4   0.230  -0.036  -0.026   -0.015   -0.015
## 707   COM_A2 ~~    C_CR4   0.230  -0.029  -0.029   -0.028   -0.028
## 708     DECO =~      Im1   0.230  -0.017  -0.021   -0.016   -0.016
## 709     DECO =~      Im2   0.229   0.015   0.019    0.015    0.015
## 710      Im1 ~~     Im16   0.229  -0.012  -0.012   -0.054   -0.054
## 711    ATMOS =~     Im19   0.229   0.023   0.029    0.025    0.025
## 712    ATMOS =~     Im16   0.229  -0.022  -0.028   -0.023   -0.023
## 713     Im10 ~~     Im22   0.226  -0.008  -0.008   -0.033   -0.033
## 714      COI =~   C_REP3   0.223  -0.005  -0.009   -0.015   -0.015
## 715      COI =~   COM_A1   0.221   0.013   0.022    0.015    0.015
## 716      Im5 ~~   COM_A1   0.220  -0.018  -0.018   -0.024   -0.024
## 717     DECO =~   COM_A1   0.220  -0.017  -0.022   -0.015   -0.015
## 718    AFCOM =~    C_CR3   0.220   0.028   0.031    0.015    0.015
## 719      Im2 ~~     Im19   0.219  -0.010  -0.010   -0.030   -0.030
## 720      COI =~     Im20   0.218  -0.013  -0.022   -0.015   -0.015
## 721     Im19 ~~    C_CR4   0.211   0.021   0.021    0.030    0.030
## 722 PRODQUAL =~      Im3   0.207   0.021   0.015    0.011    0.011
## 723   CHOICE =~    SAT_1   0.207   0.014   0.019    0.019    0.019
## 724     Im21 ~~      Im2   0.205   0.012   0.012    0.024    0.024
## 725     Im16 ~~    SAT_2   0.204   0.012   0.012    0.028    0.028
## 726     DECO =~    SAT_1   0.197  -0.013  -0.016   -0.016   -0.016
## 727      Im5 ~~    SAT_3   0.195  -0.016  -0.016   -0.022   -0.022
## 728      Im3 ~~    C_CR3   0.193  -0.015  -0.015   -0.028   -0.028
## 729      Im3 ~~   C_REP2   0.189  -0.004  -0.004   -0.045   -0.045
## 730     Im17 ~~    C_CR4   0.188   0.017   0.017    0.046    0.046
## 731      Im3 ~~    SAT_2   0.185  -0.007  -0.007   -0.027   -0.027
## 732   FRENCH =~    C_CR3   0.183  -0.029  -0.028   -0.014   -0.014
## 733      Im3 ~~   C_REP3   0.183   0.004   0.004    0.024    0.024
## 734     Im22 ~~   COM_A4   0.180   0.019   0.019    0.028    0.028
## 735    ATMOS =~    SAT_1   0.179  -0.013  -0.016   -0.017   -0.017
## 736     Im14 ~~      Im1   0.179  -0.005  -0.005   -0.064   -0.064
## 737       RI =~     Im19   0.177  -0.035  -0.021   -0.018   -0.018
## 738      Im3 ~~   COM_A1   0.174  -0.010  -0.010   -0.025   -0.025
## 739 PRODQUAL =~   C_REP1   0.171   0.014   0.010    0.014    0.014
## 740     Im14 ~~    SAT_3   0.170  -0.007  -0.007   -0.031   -0.031
## 741   C_REP3 ~~    C_CR1   0.169   0.009   0.009    0.024    0.024
## 742     DECO  ~      Im8   0.169   0.016   0.013    0.014    0.013
## 743     Im16 ~~     Im18   0.164  -0.013  -0.013   -0.022   -0.022
## 744      Im1 ~~    C_CR3   0.163  -0.015  -0.015   -0.046   -0.046
## 745 PRODQUAL  ~      SAT   0.163   0.119   0.141    0.141    0.141
## 746     Im21 ~~   C_REP2   0.160  -0.006  -0.006   -0.037   -0.037
## 747      Im3 ~~    C_CR4   0.156  -0.013  -0.013   -0.025   -0.025
## 748     Im15  ~   FRENCH   0.152   0.019   0.019    0.016    0.016
## 749    ATMOS  ~      Im8   0.151   0.017   0.014    0.014    0.014
## 750      Im1 ~~     Im19   0.151  -0.009  -0.009   -0.051   -0.051
## 751     DECO =~    C_CR3   0.150   0.020   0.025    0.012    0.012
## 752     Im19 ~~   COM_A3   0.150  -0.014  -0.014   -0.025   -0.025
## 753    SAT_3 ~~   C_REP1   0.148  -0.008  -0.008   -0.020   -0.020
## 754      Im2 ~~    C_CR4   0.147  -0.014  -0.014   -0.022   -0.022
## 755   FRENCH =~     Im18   0.144   0.016   0.016    0.011    0.011
## 756   FRENCH =~     Im17   0.144  -0.016  -0.016   -0.013   -0.013
## 757     Im17 ~~    C_CR3   0.144   0.015   0.015    0.041    0.041
## 758       RI =~    C_CR1   0.144  -0.040  -0.024   -0.012   -0.012
## 759    SAT_3 ~~   C_REP2   0.143  -0.005  -0.005   -0.035   -0.035
## 760      SAT =~   C_REP3   0.143   0.009   0.007    0.013    0.013
## 761 PRODQUAL =~     Im14   0.140   0.014   0.010    0.012    0.012
## 762 PRODQUAL =~     Im10   0.140  -0.014  -0.010   -0.011   -0.011
## 763    AFCOM =~     Im18   0.140  -0.014  -0.015   -0.011   -0.011
## 764     Im18 ~~    C_CR1   0.137   0.017   0.017    0.022    0.022
## 765     Im11 ~~    SAT_3   0.136  -0.014  -0.014   -0.019   -0.019
## 766   FRENCH =~     Im21   0.136  -0.018  -0.017   -0.013   -0.013
## 767   FRENCH =~    SAT_2   0.134  -0.014  -0.013   -0.013   -0.013
## 768     DECO  ~      Im9   0.132   0.011   0.009    0.012    0.009
## 769   C_REP1 ~~    C_CR4   0.130   0.010   0.010    0.020    0.020
## 770      SAT =~     Im19   0.125  -0.028  -0.023   -0.021   -0.021
## 771      Im4 ~~    SAT_2   0.123  -0.006  -0.006   -0.028   -0.028
## 772     Im18 ~~    SAT_2   0.122   0.008   0.008    0.020    0.020
## 773       RI =~      Im2   0.116   0.019   0.011    0.008    0.008
## 774      Im4 ~~    C_CR4   0.114   0.011   0.011    0.027    0.027
## 775      SAT =~     Im22   0.112  -0.021  -0.017   -0.011   -0.011
## 776     Im10 ~~     Im19   0.112  -0.005  -0.005   -0.024   -0.024
## 777     Im10 ~~   C_REP2   0.112  -0.002  -0.002   -0.035   -0.035
## 778      Im4 ~~   COM_A1   0.112   0.008   0.008    0.025    0.025
## 779     Im11 ~~    C_CR3   0.111   0.019   0.019    0.018    0.018
## 780     DECO =~     Im19   0.109   0.021   0.026    0.023    0.023
## 781     DECO =~     Im16   0.109  -0.020  -0.025   -0.021   -0.021
## 782      Im4 ~~     Im14   0.107  -0.003  -0.003   -0.035   -0.035
## 783     Im17 ~~    SAT_1   0.105   0.006   0.006    0.039    0.039
## 784   COM_A2 ~~   C_REP3   0.104  -0.006  -0.006   -0.017   -0.017
## 785     Im21 ~~    SAT_1   0.104   0.009   0.009    0.020    0.020
## 786     Im21 ~~     Im12   0.104   0.009   0.009    0.020    0.020
## 787     Im18 ~~    C_CR4   0.102  -0.015  -0.015   -0.018   -0.018
## 788     PROF =~    C_CR3   0.101  -0.023  -0.022   -0.011   -0.011
## 789   CHOICE =~      Im6   0.099  -0.009  -0.012   -0.010   -0.010
## 790   CHOICE =~      Im7   0.099   0.011   0.014    0.011    0.011
## 791      Im3 ~~    SAT_3   0.098  -0.007  -0.007   -0.018   -0.018
## 792     Im20 ~~    C_CR4   0.098  -0.018  -0.018   -0.019   -0.019
## 793      SAT =~   C_REP2   0.096  -0.007  -0.006   -0.009   -0.009
## 794      SAT =~      Im7   0.096  -0.019  -0.016   -0.013   -0.013
## 795     Im16 ~~   C_REP2   0.094  -0.004  -0.004   -0.030   -0.030
## 796     PROF =~    SAT_3   0.093  -0.019  -0.018   -0.016   -0.016
## 797      Im5 ~~     Im12   0.092   0.009   0.009    0.018    0.018
## 798      SAT =~      Im3   0.091   0.010   0.009    0.007    0.007
## 799      Im6 ~~    SAT_2   0.089  -0.007  -0.007   -0.017   -0.017
## 800       RI =~    C_CR3   0.089   0.033   0.020    0.009    0.009
## 801     Im20 ~~     Im11   0.085   0.011   0.011    0.016    0.016
## 802      Im2 ~~      Im6   0.085  -0.006  -0.006   -0.015   -0.015
## 803     Im14 ~~   COM_A1   0.084   0.005   0.005    0.022    0.022
## 804     Im21 ~~    C_CR1   0.082   0.015   0.015    0.017    0.017
## 805      Im5 ~~   C_REP3   0.082   0.004   0.004    0.014    0.014
## 806      Im2 ~~    C_CR3   0.081   0.011   0.011    0.016    0.016
## 807      Im5 ~~     Im18   0.081   0.009   0.009    0.014    0.014
## 808      Im4 ~~     Im19   0.078  -0.005  -0.005   -0.025   -0.025
## 809    AFCOM =~     Im22   0.077  -0.015  -0.016   -0.010   -0.010
## 810     FOOD =~     Im17   0.073   0.014   0.011    0.009    0.009
## 811     FOOD =~     Im18   0.073  -0.014  -0.011   -0.008   -0.008
## 812      COI =~      Im6   0.073  -0.006  -0.010   -0.008   -0.008
## 813   COM_A2 ~~    C_CR3   0.072  -0.017  -0.017   -0.016   -0.016
## 814    AFCOM =~      Im3   0.072  -0.007  -0.007   -0.006   -0.006
## 815      Im4 ~~     Im21   0.071   0.006   0.006    0.020    0.020
## 816      Im8  ~    BRAND   0.071  -0.009  -0.011   -0.011   -0.011
## 817      Im7 ~~    C_CR3   0.070  -0.012  -0.012   -0.030   -0.030
## 818     Im11 ~~    C_CR1   0.069  -0.014  -0.014   -0.015   -0.015
## 819      Im4 ~~    SAT_1   0.068  -0.004  -0.004   -0.023   -0.023
## 820     Im18 ~~    C_CR3   0.067  -0.013  -0.013   -0.015   -0.015
## 821     Im11 ~~     Im16   0.067  -0.010  -0.010   -0.014   -0.014
## 822     Im12 ~~     Im18   0.067  -0.006  -0.006   -0.016   -0.016
## 823     Im19 ~~   C_REP3   0.065   0.003   0.003    0.015    0.015
## 824      Im4 ~~    SAT_3   0.062   0.005   0.005    0.018    0.018
## 825    BRAND =~   C_REP2   0.060   0.004   0.005    0.008    0.008
## 826      Im7 ~~    SAT_1   0.059  -0.005  -0.005   -0.031   -0.031
## 827     Im22 ~~   COM_A2   0.057   0.010   0.010    0.015    0.015
## 828     Im20 ~~     Im12   0.053   0.007   0.007    0.015    0.015
## 829   CHOICE =~    SAT_3   0.051   0.009   0.011    0.010    0.010
## 830    SAT_3 ~~    C_CR4   0.051   0.013   0.013    0.013    0.013
## 831   FRENCH =~    SAT_1   0.051  -0.009  -0.008   -0.008   -0.008
## 832    BRAND =~    C_CR3   0.049  -0.012  -0.014   -0.007   -0.007
## 833      Im3 ~~     Im13   0.049   0.004   0.004    0.015    0.015
## 834     Im20 ~~    C_CR1   0.048   0.012   0.012    0.014    0.014
## 835     FOOD =~    C_CR3   0.048  -0.017  -0.014   -0.007   -0.007
## 836      Im2 ~~    SAT_3   0.047  -0.006  -0.006   -0.012   -0.012
## 837      COI =~      Im7   0.046  -0.006  -0.009   -0.008   -0.008
## 838      Im4 ~~     Im20   0.046  -0.005  -0.005   -0.017   -0.017
## 839   CHOICE  ~      Im9   0.044  -0.007  -0.005   -0.007   -0.005
## 840      Im3 ~~     Im16   0.044  -0.004  -0.004   -0.013   -0.013
## 841     Im10 ~~    SAT_1   0.043   0.003   0.003    0.015    0.015
## 842     FOOD =~      Im3   0.043  -0.007  -0.006   -0.005   -0.005
## 843     Im11 ~~   C_REP3   0.041   0.003   0.003    0.010    0.010
## 844     PROF  ~      Im9   0.040  -0.005  -0.005   -0.007   -0.005
## 845     Im21 ~~     Im13   0.039  -0.006  -0.006   -0.012   -0.012
## 846    SAT_2 ~~    C_CR4   0.037   0.008   0.008    0.012    0.012
## 847      Im1 ~~   C_REP1   0.037  -0.002  -0.002   -0.019   -0.019
## 848     Im19 ~~      Im6   0.036  -0.005  -0.005   -0.012   -0.012
## 849      Im5 ~~     Im10   0.034  -0.003  -0.003   -0.011   -0.011
## 850      Im4 ~~   C_REP1   0.032  -0.002  -0.002   -0.013   -0.013
## 851      Im1 ~~    C_CR4   0.032  -0.007  -0.007   -0.020   -0.020
## 852     Im22 ~~    C_CR3   0.029   0.010   0.010    0.011    0.011
## 853    AFCOM =~    SAT_3   0.029  -0.008  -0.009   -0.008   -0.008
## 854     Im18 ~~   COM_A4   0.028  -0.006  -0.006   -0.010   -0.010
## 855    SAT_1 ~~    SAT_2   0.028   0.009   0.009    0.030    0.030
## 856      Im2 ~~   COM_A2   0.028  -0.005  -0.005   -0.009   -0.009
## 857   COM_A2 ~~   C_REP1   0.028   0.004   0.004    0.009    0.009
## 858      Im6 ~~   C_REP1   0.025  -0.003  -0.003   -0.008   -0.008
## 859     Im20 ~~     Im16   0.024   0.006   0.006    0.009    0.009
## 860    BRAND =~     Im21   0.024  -0.007  -0.008   -0.006   -0.006
## 861      Im5 ~~    C_CR1   0.023   0.008   0.008    0.009    0.009
## 862     Im21 ~~     Im19   0.023  -0.005  -0.005   -0.009   -0.009
## 863      COI =~     Im10   0.023   0.002   0.003    0.003    0.003
## 864     Im19 ~~    C_CR1   0.023   0.007   0.007    0.011    0.011
## 865     Im18 ~~   COM_A3   0.022   0.005   0.005    0.008    0.008
## 866     DECO =~     Im21   0.019  -0.006  -0.007   -0.005   -0.005
## 867      Im3 ~~     Im14   0.019  -0.001  -0.001   -0.012   -0.012
## 868     Im13 ~~    C_CR1   0.019  -0.006  -0.006   -0.009   -0.009
## 869      Im3 ~~     Im21   0.018  -0.003  -0.003   -0.008   -0.008
## 870     Im14 ~~     Im19   0.018   0.002   0.002    0.012    0.012
## 871       RI =~      Im3   0.018   0.006   0.004    0.003    0.003
## 872     Im22 ~~    C_CR4   0.018   0.007   0.007    0.009    0.009
## 873      Im3 ~~   C_REP1   0.017   0.001   0.001    0.007    0.007
## 874     Im13 ~~    SAT_1   0.016  -0.003  -0.003   -0.009   -0.009
## 875      Im3 ~~      Im6   0.016   0.002   0.002    0.007    0.007
## 876 PRODQUAL =~    C_CR3   0.015  -0.012  -0.008   -0.004   -0.004
## 877      Im3 ~~     Im10   0.015   0.001   0.001    0.008    0.008
## 878    AFCOM =~    C_CR1   0.014   0.007   0.007    0.004    0.004
## 879      COI =~     Im14   0.014  -0.001  -0.002   -0.003   -0.003
## 880 PRODQUAL  ~      Im8   0.013   0.003   0.004    0.004    0.004
## 881   COM_A2 ~~   C_REP2   0.013   0.002   0.002    0.011    0.011
## 882     PROF =~     Im10   0.012   0.004   0.003    0.004    0.004
## 883     PROF =~     Im14   0.012  -0.004  -0.003   -0.004   -0.004
## 884   C_REP2 ~~    C_CR1   0.012   0.002   0.002    0.011    0.011
## 885 PRODQUAL =~    SAT_3   0.011   0.007   0.005    0.005    0.005
## 886     Im10 ~~      Im1   0.011  -0.001  -0.001   -0.013   -0.013
## 887     Im21 ~~   COM_A3   0.011  -0.004  -0.004   -0.006   -0.006
## 888     Im16 ~~   COM_A2   0.010   0.004   0.004    0.006    0.006
## 889       RI =~    SAT_3   0.010   0.007   0.004    0.004    0.004
## 890     FOOD =~   C_REP3   0.009  -0.002  -0.002   -0.003   -0.003
## 891       RI =~    C_CR4   0.009   0.011   0.006    0.003    0.003
## 892     Im16 ~~      Im6   0.009  -0.003  -0.003   -0.005   -0.005
## 893     PROF ~~    AFCOM   0.009  -0.012  -0.014   -0.014   -0.014
## 894    AFCOM =~     Im13   0.008  -0.003  -0.003   -0.003   -0.003
## 895      Im6 ~~   COM_A4   0.007  -0.003  -0.003   -0.005   -0.005
## 896     Im21 ~~      Im6   0.007  -0.003  -0.003   -0.004   -0.004
## 897       RI =~     Im20   0.007   0.007   0.004    0.003    0.003
## 898      Im4 ~~     Im10   0.006  -0.001  -0.001   -0.007   -0.007
## 899   FRENCH =~   C_REP3   0.006   0.001   0.001    0.003    0.003
## 900      COI =~     Im18   0.006   0.002   0.003    0.002    0.002
## 901    ATMOS =~      Im7   0.005  -0.003  -0.004   -0.003   -0.003
## 902    ATMOS =~      Im6   0.005   0.002   0.003    0.003    0.003
## 903     FOOD  ~      SAT   0.003   0.019   0.019    0.019    0.019
## 904     Im18 ~~    SAT_3   0.003   0.002   0.002    0.003    0.003
## 905      Im5 ~~   COM_A3   0.003   0.002   0.002    0.003    0.003
## 906     FOOD =~     Im12   0.003  -0.003  -0.002   -0.002   -0.002
## 907   FRENCH =~     Im10   0.003   0.002   0.002    0.002    0.002
## 908   FRENCH =~     Im14   0.003  -0.002  -0.002   -0.002   -0.002
## 909     Im13 ~~    C_CR4   0.002   0.002   0.002    0.003    0.003
## 910     DECO =~     Im14   0.002   0.001   0.001    0.001    0.001
## 911     DECO =~     Im10   0.002  -0.001  -0.001   -0.001   -0.001
## 912 PRODQUAL =~     Im21   0.001   0.003   0.002    0.001    0.001
## 913   C_REP1 ~~    C_CR1   0.001  -0.001  -0.001   -0.002   -0.002
## 914     Im12 ~~      Im1   0.001   0.001   0.001    0.005    0.005
## 915      Im4 ~~     Im13   0.001  -0.001  -0.001   -0.003   -0.003
## 916   FRENCH =~      Im4   0.001   0.001   0.001    0.001    0.001
## 917   COM_A3 ~~    C_CR4   0.001  -0.002  -0.002   -0.002   -0.002
## 918   COM_A4 ~~   C_REP2   0.001   0.001   0.001    0.003    0.003
## 919      COI =~     Im21   0.001   0.001   0.002    0.001    0.001
## 920    BRAND =~   COM_A1   0.001  -0.001  -0.002   -0.001   -0.001
## 921   FRENCH =~     Im12   0.001  -0.001  -0.001   -0.001   -0.001
## 922      Im5 ~~     Im13   0.001  -0.001  -0.001   -0.002   -0.002
## 923    SAT_1 ~~   C_REP1   0.001   0.000   0.000   -0.002   -0.002
## 924     Im12 ~~     Im19   0.001   0.001   0.001    0.002    0.002
## 925     Im14 ~~   COM_A4   0.001   0.001   0.001    0.002    0.002
## 926      COI =~     Im17   0.001   0.001   0.001    0.001    0.001
## 927      Im1 ~~      Im7   0.001   0.001   0.001    0.005    0.005
## 928     Im11 ~~     Im19   0.001   0.001   0.001    0.001    0.001
## 929     Im13 ~~   C_REP3   0.000   0.000   0.000   -0.001   -0.001
## 930   COM_A3 ~~    C_CR1   0.000  -0.001  -0.001   -0.001   -0.001
## 931      Im5 ~~     Im20   0.000  -0.001  -0.001   -0.001   -0.001
## 932     Im16 ~~     Im17   0.000   0.000   0.000   -0.001   -0.001
## 933     Im19 ~~   COM_A4   0.000   0.000   0.000    0.001    0.001
## 934     PROF =~   C_REP2   0.000   0.000   0.000    0.000    0.000
## 935     Im10 ~~   C_REP1   0.000   0.000   0.000    0.000    0.000
## 936      Im6 ~~    C_CR3   0.000   0.000   0.000    0.000    0.000
## 937     DECO =~   COM_A4   0.000   0.000   0.000    0.000    0.000